Structural illumination and evanescent coupling for the extension of imaging interferometric microscopy

ABSTRACT

In accordance with the aspects of the present disclosure, a method and apparatus is disclosed for three-dimensional imaging interferometric microscopy (IIM), which can use at least two wavelengths to image a three-dimensional object. The apparatus can include a first, a second, and a third optical system. The first optical system is disposed to provide a substantially coherent illumination to the 3D object, wherein the illumination is characterized by a plurality of wavelengths. The second optical system includes an optical image recording device and one or more additional optical components characterized by a numerical aperture NA. The third optical system provides interferometric reintroduction of a portion of the coherent illumination as a reference beam into the second optical system. An image recording device records each sub-image formed as a result of interference between the illumination that is scattered by the 3D object and the reference beam.

RELATED APPLICATIONS

This application is a continuation-in-part to U.S. patent applicationSer. No. 13/629,598 filed on Sep. 27, 2012, which is acontinuation-in-part to U.S. patent application Ser. No. 13/345,267filed on Jan. 6, 2012, now U.S. Pat. No. 8,526,105 issued on Sep. 3,2012, which is a divisional of U.S. patent application Ser. No.12/347,619 filed Dec. 31, 2008, now U.S. Pat. No. 8,115,992 issued onFeb. 14, 2012, and claims priority from U.S. Provisional PatentApplication Ser. Nos. 61/017,985, filed Dec. 31, 2007; 61/089,669, filedAug. 18, 2008; and 61/115,246, filed Nov. 17, 2008, which are herebyincorporated by reference in their entirety.

GOVERNMENT RIGHTS

This invention was made with government support under Contract Nos.HR0011-05-1-0006 awarded by the Defense Advanced Research ProjectsAgency and FA9550-06-1-0001 awarded by the Air Force Office ofScientific Research. The government has certain rights in the invention.

FIELD OF THE INVENTION

This invention relates generally to microscopy, and, more particularly,to an imaging interferometric microscope.

BACKGROUND OF THE INVENTION

Optical microscopy is among the oldest applications of optical scienceand remains one of the most widely used optical technologies. In spiteof impressive results obtained by fluorescent microscopy in exceedingthe classical diffraction limit, non-fluorescent transmission/reflectionmicroscopy remains an important field of modern research. However, usingtraditional illumination schemes, resolution is limited to ˜K₁λ/NA whereλ is the source wavelength and NA is the numerical aperture (sine of thehalf-acceptance angle) of the imaging objective lens. The “constant” K₁depends on both the details of the image and on the illumination scheme.For example, K₁ can be between 0.25 and 1. Hence, traditional approachesto improve resolution are either to use shorter wavelengths and/or touse larger numerical-aperture lenses. For biological samples, however,the wavelength is constrained to the visible spectral range becauseultraviolet photons can damage samples. In many practical cases, evenfor inorganic samples, the wavelength is limited to the deep ultraviolet(for example 193 nm) since transmissive optical materials becomedifficult at shorter wavelengths (fused quartz has a cutoff at ˜185 nm).Furthermore, a disadvantage of using a high-NA lens is the resultingshort depth-of-field (an essential feature of achieving high resolutionin a single image; typically the depth-of-field scales as K₂λ/NA² whereK₂ is a second “constant” of order unity). The depth-of-field decreasesrapidly as the NA is increased to increase the resolution. In addition,the field of view (the area over which the resolution is achieved) andthe working distance (the distance from the final lens surface to theobject plane) are reduced for higher-NA optical systems. These lattertwo issues can be surmounted by more complex objective lenses, with anincrease in the cost of manufacturing. These tradeoffs are well knownand are discussed in many microscopy overviews.

Synthetic aperture approaches, such as, for example, imaginginterferometric microscopy (IIM), extend the collected spatialfrequencies to improve the image. IIM, with both illumination andcollection in a transmission medium (usually air), uses a low-NAobjective and yet provides a resolution approximately a factor of twobetter than that available even with a high-NA objective usingconventional coherent or incoherent illumination. A major advantage isthat the depth-of-field, field-of-view and working distance associatedwith the low-NA system are retained, but the final composite image has aresolution at the linear system limit imposed by the transmission medium(≧λ/4n where λ is the wavelength in free space and n is the index ofrefraction of the transmission medium), and significantly better thanthat accessible with even a high NA lens using conventional (coherent orincoherent) illumination approaches.

An exemplary IIM with two offset partial images, one each in orthogonalspatial directions can result in an increased resolution by three timesusing about 0.4-NA objective and 633-nm He—Ne laser. Furthermore, IIMrequires building an interferometric system around the objective lenswhich is an issue for wide-spread adoption of this approach, and inparticular towards its adoption to the existing microscopes. In theprior art, this interferometer required additional optics to relay thepupil plane of the collection objective to convenient location; this isstraightforward but required significant additional optics. Hence, thereis a need for a new approach that does not require a large change to theimaging optical system that comprises the objective lens and subsequentoptical components.

The prior art imaging interferometric microscopy was able to imagemaximum spatial frequency of 2λ/λ e.g. to the linear system's limit ofthe air (transmission medium between the object and the lens). Theultimate linear system limit is 2πn/λ, which reflects the use of animmersion medium of refractive index n. Even though materials withrefractive indices of up to about 5 are known at some opticalwavelengths, the highest numerical aperture available for the immersionmicroscopy is about 1.4, limited by the refractive index of the glassused to make the lens, by the refractive indices available for the indexmatching fluids, and the well known difficulties of making aberrationcorrected optics of high NA. Hence, there is a need for a new approachthat can achieve this linear system limit without requiringindex-matching fluids or high NA lenses.

As is well-known, using off-axis illumination provides enhancedresolution over that available with either of the standard illuminationschemes discussed above, but there is some distortion of the imageassociated with the resultant non-constant transfer function fordifferent regions of frequency space. This non-uniform frequency-spacecoverage can be addressed with appropriate pupil plane filters and bycombining partial images corresponding to different parts of frequencyspace, as has been previously demonstrated in the case of imaginginterferometric lithography

SUMMARY OF THE INVENTION

In implementations, a method for imaging a 3D object immersed in amedium of index of refraction n_(med) is disclosed. The method caninclude providing a first optical system disposed to provide asubstantially coherent illumination to the 3D object, wherein theillumination is characterized by a plurality of wavelengths λ^(j), j=1,2, . . . m, with λ^(j+1)<λ^(j), wherein the plurality of wavelengthsspan a wavelength range of Δλ=λ¹-λ^(m); at each λ^(j) the illuminationis characterized by a center position, a radius of curvature, auniform-intensity illumination diameter at a plane of the 3D object, anda wavevector wherein the wavevector is disposed at one of a plurality ofincident wavevectors from about 0 to about 2λn_(med)/λ^(j), with respectto a longitudinal axis of the 3D object and at a plurality of azimuthangles spanning about 0 to 2π; providing a second optical systemcomprising an optical image recording device and one or more additionaloptical components with a numerical aperture NA, the second opticalsystem defining an optical axis, wherein the optical recording device isoperable to collect at least a portion of the illumination from thefirst optical system scattered from the 3D object, wherein the opticalaxis of the second optical system is disposed at one of a plurality ofangles between 0 and π/2 with respect to the longitudinal axis of the 3Dobject and at a plurality of azimuth angles spanning about 0 to 2π,wherein the field-of-view of the second optical system is within aspatial extent of the uniform-intensity illumination provided by thefirst optical system; providing a third optical system disposed in anoptical path of the first optical system to provide interferometricreintroduction of a portion of the coherent illumination at each λ^(j)as a reference beam into the second optical system, wherein each of anamplitude, a phase, a radius of curvature, a path length, and an angleof incidence of the reference beam is adjustable such that a referenceillumination suitable for interfering with a portion of the illuminationscattered by the 3D object and collected by the second optical system ispresent at an input of the optical image recording device; recording aplurality of sub-images of the 3D object at the optical image recordingdevice, one at each λ^(j), wherein each sub-image is formed as a resultof interference between scattering resulting from the coherentillumination of the 3D object and the reference beam; adjusting thefirst, the second and the third optical systems to collect a pluralityof sub-images corresponding to the plurality of wavelengths, to aplurality of off-axis illumination conditions, and additionally to aplurality of directions of the optical axis of the second optical systemwith respect to the longitudinal axis of the 3D object; combining theplurality of sub-images into a separate composite images of the 3Dobject.

In implementations, the method can further include translating a centerof a field-of-view of the second optical system relative to a centerposition of an illumination spatial extent provided by the first opticalsystem, to extend an area of the 3D image.

In implementations, the 3D object can include two substantially 2Dobjects separated from each other with a plane-parallel-boundedhomogenous medium characterized by a thickness and an index ofrefraction and wherein the plurality of wavelengths is reduced to two,λ¹ and λ², and the longitudinal axis is defined as a normal to theplane-parallel-bounded homogenous medium.

In implementations, the method further can further include providing abody composed of a homogeneous medium of index of refraction n_(pp)greater than n_(med) within which the 3D object is immersed and having aplane exit face as a final surface of the first optical system; locatingthe 3D object at a distance less than λ_(avg) from the plane exit faceof the body; providing for coupling of the coherent illumination to thebody by one of side-coupling, prism coupling an addition of a grating toa face of the body opposite the exit face; and whereby the illuminationprovided by the first optical system is at a wavevector larger than2πn_(med)/λ^(j) and less than 2πn_(pp)/λ^(j) and is an evanescent waveextending from the plane exit face of the body.

In implementations, the method can further include providing aplane-parallel-bounded body composed of a homogeneous medium of index ofrefraction n_(pp) greater than n_(med) and a plane exit face as a finalelement of the first optical system; providing for coupling of thecoherent illumination to the body by addition of a grating to the faceof the plane-parallel-bounded body opposite the exit face; locating the3D object at a distance less than λ_(avg) from the plane exit face ofthe plane-parallel body; whereby the illumination provided by the firstoptical system is at a wavevector larger than 2πn_(med)/λ^(j) and lessthan 2πn_(pp)/λ^(j) and is an evanescent wave extending from the planeexit face of the plane-parallel body; adjusting the second opticalsystem to collect illumination scattered by the 3D object from theillumination provided by the first optical system wherein theillumination that is scattered by the 3D object is at a wavevectorbetween 2πn_(med)/λ^(j) and 2πn_(pp)/λ^(j) and is evanescently coupledinto the plane-parallel-bounded body and is coupled out of theplane-parallel-bounded body by a grating on the plane exit face of theplane-parallel-bounded body opposite the 3D object.

In implementations, providing the third optical system can furtherinclude collecting a portion of the coherent illumination at each λ^(j)by splitting the coherent illumination using a beam splitter disposed inan optical path of the first optical system, and interferometricallyreintroducing the portion of the coherent illumination as a referencebeam after an exit aperture of a collection lens of the second opticalsystem, wherein the reintroduction is at one of a position, anamplitude, a phase, a radius of curvature, a path length, and an angleof incidence into the third optical system such that a sub-image isformed with spatial frequency content that is directly related to aspatial frequency content of the illumination that is scattered by the3D object.

In implementations, providing the third optical system can furtherinclude collecting a portion of the coherent illumination at each λ^(j)by splitting the coherent illumination using a first beam combiningdevice disposed in an optical path of the first optical system, andinterferometrically reintroducing the portion of the coherentillumination as a reference beam before an entrance aperture of acollection lens of the second optical system, wherein the reintroductionis at an angle less than sin⁻¹(NA) of the collection lens, wherein thesecond beam combining device is selected from a group consisting of: abeamsplitter, a grating coupler, and a waveguide filter such that asub-image is formed with spatial frequency content that is directlyrelated to a spatial frequency content of the illumination that isscattered by the 3D object.

In implementations, the method can further include obtaining additionalsub-images by adjusting the phase of the reference beam provided by thethird optical system at the optical image recording device relative to aphase of the illumination provided by the first optical system at the 3Dobject.

In implementations, the method can further include comprisingcomputationally manipulating each of the sub-images to correct fordistortions, spatial frequency aliasing, and alterations introduced byarrangements of the first, second, and third optical systems.

In implementations, the Illumination can include combinations of twowavelengths (λ^(j) and λ^(j′)) and the method can further includedetecting at an anti-Stokes wavelength [λ^(j)λ^(j′)/(2λ^(j)−λ^(j′))] andtuning a difference between the two wavelengths to obtain a coherentanti-Stokes Raman signature of the 3D object.

In implementations, an apparatus for imaging a 3D object immersed in amedium of index of refraction n_(med) with a thickness larger thanoptical wavelengths in the medium used for the imaging is disclosed. Theapparatus can include a mechanical mechanism to support the 3D object; afirst optical system disposed to provide a substantially coherentillumination to the 3D object, wherein the illumination is characterizedby a plurality of wavelengths λ^(j), j=1, 2, . . . m, withλ^(j+1)<λ^(j), wherein the plurality of wavelengths span a wavelengthrange of Δλ=λ¹-λ^(m); at each λ^(j) the illumination is characterized bya center position, a radius of curvature, an uniform-intensityillumination diameter at a plane of the 3D object, and a wavevectorwherein the wavevector is disposed at one of a plurality of incidentwavevectors from about 0 to about 2πn_(med)/λ^(j), with respect to alongitudinal axis of the 3D object and at a plurality of azimuth anglesspanning about 0 to 2π; a second optical system comprising an opticalimage recording device and one or more additional optical componentscharacterized by a numerical aperture NA, the second optical systemdefining an optical axis, wherein the optical recording device isoperable to collect at least a portion of the illumination from thefirst optical system scattered from 3D object, wherein the optical axisof the second optical system is disposed at one of a plurality of anglesbetween 0 and π/2 with respect to the longitudinal axis of the objectand at a plurality of azimuthal angles spanning about 0 to 2π, whereinthe field-of-view of the second optical system is within a spatialextent of the uniform-intensity illumination provided by the firstoptical system; a third optical system disposed in an optical path ofthe first optical system to provide interferometric reintroduction of aportion of the coherent illumination at each λ^(j) as a reference beaminto the second optical system, wherein each of an amplitude, a phase, aradius of curvature, a path length, and an angle of incidence of thereference beam is adjustable such that a reference illumination suitablefor interfering with the portion of the illumination scattered by the 3Dobject and collected by the second optical system is present at an inputof the optical image recording device; the image recording devicewherein each sub-image formed as a result of interference between theillumination that is scattered by the 3D object and the reference beamat each λ^(j) is recorded; an adjustment mechanism operable to configurethe first, the second, and the third optical systems to collect aplurality of sub-images corresponding to the plurality of wavelengths,to a plurality of illumination and additionally to a plurality ofregions of an object spatial frequency space; and a signal-processingdevice operable to combine the plurality of sub-images into a separatecomposite image of the 3D object.

In implementations, the apparatus can further include one or moreoptical, mechanical or both optical and mechanical elements operable totranslate a center of a field-of-view of the second optical systemrelative to a center position of an illumination spatial extent providedby the first optical system, to extend an area of the 3D image.

In implementations, the 3D object can include two substantially 2Dobjects separated from each other with a plane-parallel-boundedhomogenous medium characterized by a thickness and an index ofrefraction and wherein the plurality of wavelengths is reduced to two,λ¹ and λ², and the longitudinal axis is defined as the normal to theplane-parallel-bounded homogenous medium.

In implementations, the apparatus can further include a body composed ofa homogeneous medium of index of refraction n_(pp) greater than n_(med)and having a plane exit face as a final surface of the first opticalsystem; a coupling element operable to couple the coherent illuminationto the body by one of side-coupling, prism coupling or an addition of agrating to a face of the body; wherein the 3D object is positionable ata distance less than λ_(avg) from the plane exit face of the body;whereby the illumination provided by the first optical system is at awavevector larger than 2πn_(med)/λ^(j) and less than 2πn_(pp)/λ^(j) andis an evanescent wave extending from the plane exit face of the body.

In implementations, the apparatus can further include aplane-parallel-bounded body composed of a homogeneous medium of index ofrefraction n_(pp) greater than n_(med) and a plane exit face as a finalelement of the first optical system; wherein the 3D object ispositionable at a distance less than λ_(avg) from the plane exit face ofthe body; a coupling element operable to couple the coherentillumination into the body by addition of a grating to a face of theplane-parallel-bounded body opposite the exit face; whereby theillumination provided by the first optical system is at a wavevectorlarger than 2πn_(med)/λ^(j) and less than 2πn_(pp)/λ^(j) and is anevanescent wave extending from the plane exit face of the body; anadjustment element operable to adjust the second optical system tocollect light scattered by the 3D object from the illumination providedby the first optical system wherein the illumination that is scatteredby the 3D object is at a wavevector between 2πn_(med)/λ^(j) and2πn_(pp)/λ^(j), is evanescently coupled into the plane-parallel-boundedbody and is coupled out of the plane-parallel-bounded body by a gratingon the plane exit face of the plane-parallel-bounded body opposite the3D object.

In implementations, the third optical system can further be operable tocollect a portion of the coherent illumination at each λ^(j) bysplitting the coherent illumination using a beam splitter disposed in anoptical path of the first optical system, and interferometricallyreintroduce the portion of the coherent illumination as a reference beamafter an exit aperture of a collection lens of the second opticalsystem, wherein the reintroduction is at one of a position, anamplitude, a phase, a radius of curvature, a path length, and an angleof incidence into the third optical system such that a sub-image isformed with spatial frequency content that is directly related to thespatial frequency content of the illumination that is scattered by the3D object.

In implementations, Claim 11, the third optical system can further beoperable to collect a portion of the coherent illumination at each λ^(j)by splitting the coherent illumination using a first beam combiningdevice disposed in an optical path of the first optical system, andinterferometrically reintroduce the portion of the coherent illuminationas a reference beam before an entrance aperture of a collection lens ofthe second optical system, wherein the reintroduction is at an angleless than the sin⁻¹(NA) of the collection lens, wherein the second beamcombining device is selected from a group consisting of: a beamsplitter,a grating coupler, and a waveguide filter such that a sub-image isformed on the optical image recording device with spatial frequenciesdirectly related to spatial frequency content of the illumination thatis scattered by the 3D object.

In implementations, additional sub-images can be obtained by adjusting aphase of the reference beam provided by the third optical system at theoptical image recording device relative to a phase of the illuminationbeam provided by the first optical system at the 3D object.

In implementations, the apparatus can further include a signalprocessing unit comprising a processor and a memory storing one or morealgorithms that cause the processor to computationally manipulating eachof the sub-Images to correct for distortions, spatial frequencyaliasing, and alterations introduced by the combinations of the first,second, and third optical systems.

In implementations, the first optical system can be operable to provideillumination with combinations of two wavelengths (λ^(j) and λ^(j′)) andthe signal processing unit is for operable to detect at an anti-Stokeswavelength [λ_(j)λ^(j′)/(2λ^(j)−λ^(j′))] and tune the difference betweenthe two wavelengths to obtain a spatially resolved coherent anti-StokesRaman signature of the 3D object.

Additional objects and advantages of the Invention will be set forth inpart in the description which follows, and in part will be obvious fromthe description, or may be learned by practice of the invention. Theobjects and advantages of the invention will be realized and attained bymeans of the elements and combinations particularly pointed out in theappended claims.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory onlyand are not restrictive of the invention, as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of this specification, illustrate several embodiments of theinvention and together with the description, serve to explain theprinciples of the invention.

FIG. 1 shows an exemplary prior art imaging interferometric microscopy(IIM) experimental arrangement.

FIG. 2A shows the frequency space coverage for conventional normalincidence coherent illumination.

FIG. 2B shows the frequency space coverage for an off-axis incidencecoherent illumination.

FIG. 3 shows an exemplary structured illumination approach to IIM,according to various embodiments of the present teachings.

FIG. 4 shows the schematic of structural illumination and restorationalgorithms, in accordance with the present teachings.

FIG. 5A is a schematic illustration showing a dynamic physical block ina back pupil plane of the second optical system to alternately block andunblock the reference beam, according to present teachings.

FIG. 5B is a schematic illustration showing injection of the referencebeam into a second optical system using a prism, according to presentteachings.

FIG. 5C is a schematic illustration of injection of the reference beaminto a second optical system using a beamsplitter, according to presentteachings.

FIG. 5D is a schematic illustration showing blocking of the referencebeam with a k-vector filter, according to present teachings.

FIG. 5E is a schematic illustration showing injection of the referencebeam with a grating, according to present teachings.

FIG. 6 shows k-vector filter characteristic of an exemplary SiN-on-glassguided mode resonance filter.

FIG. 7A shows the frequency space coverage for the arrangement of FIG.3, with intermediate frequency offset within bandpass of lens.

FIG. 7B shows the frequency space coverage for the arrangement of FIG. 3after removal of the dark field and intermediate frequency imaging termsand correction of the observed frequencies.

FIG. 8A schematically illustrates a Manhattan geometry pattern used forimage resolution exploration consisting of five nested “ells” and alarge box.

FIG. 8B illustrates intensity Fourier space components of the Manhattangeometry pattern shown in FIG. 8A, mapped onto a frequency spacecoverage of the imaging system.

FIGS. 9A-9F show the preliminary results of an experiment using anNA=0.4 objective with a He—Ne laser illumination (λ=633 nm) and withabout 240 nm structure with corresponding simulations using theconfiguration presented in FIG. 5A.

FIG. 10A shows reconstructed images of 260 nm and 240 nm CD structuresobtained using the optical configuration of FIG. 5A after the dark fieldsubtraction, frequency shifting correction, and sub-image combination.

FIG. 10B show a crosscut (gray) of the images of FIG. 10A compared witha crosscut of corresponding simulation (black).

FIG. 11A shows reconstructed images of 260 nm and 240 nm CD structuresobtained using the optical arrangement shown in FIG. 5E.

FIG. 11B shows a crosscut (gray) of the images of FIG. 10A compared witha crosscut of corresponding simulation (black).

FIG. 12A shows an exemplary IIM arrangement with evanescentillumination, according to various embodiments of the present teachings.

FIG. 12B shows an exemplary IIM arrangement with evanescent illuminationwith a rotated optical axis, according to various embodiments of thepresent teachings.

FIGS. 13A-13C show alternate approaches for coupling light for substrateillumination, in accordance with various embodiments.

FIG. 14A schematically illustrates a Manhattan geometry pattern used forimage resolution exploration consisting of five nested “ells” and alarge box.

FIG. 14B illustrates intensity Fourier space components of the Manhattangeometry pattern shown in FIG. 14A, mapped onto a frequency spacecoverage of the imaging system, for a structure with CD=180 nm for theoptical arrangement shown in FIG. 3.

FIG. 14C illustrates intensity Fourier space components of the Manhattangeometry pattern shown in FIG. 12A, mapped onto a frequency spacecoverage of the imaging system, for a structure with CD=150 nm for theoptical arrangement shown in FIG. 11A.

FIG. 15A show reconstructed image of 260 nm and 240 nm CD structuresobtained using the optical arrangement shown in FIG. 12A.

FIG. 15B show a crosscut (gray) of the images of FIG. 15A compared witha crosscut of corresponding simulation (black).

FIG. 16A shows a reconstructed high frequency image of a 150 nmstructure using evanescent illumination and a tilted optical system,shown in FIG. 12B.

FIG. 16B shows a high frequency image simulation of a 150 nm structureusing evanescent illumination and a tilted optical system, shown in FIG.12B.

FIG. 16C shows experimental and simulation cross-cuts of images shown inFIGS. 16A and 168.

FIG. 16D shows a reconstructed composite image of a 150 nm structureusing evanescent illumination and a tilted optical system, shown in FIG.12B

FIG. 16E shows a composite image simulation of a 150 nm structure usingevanescent illumination and a tilted optical system, shown in FIG. 12B.

FIG. 16F shows experimental and simulation cross-cuts of images shown inFIGS. 16D and 16E.

FIG. 17 shows available frequency space coverage for various IIM opticalconfigurations, in accordance with the present teachings.

FIG. 18 shows a schematic diagram showing the high angle light scatteredfrom an object and extracted from the substrate using at least onegrating, in accordance with various embodiments of the presentteachings.

FIG. 19 shows prism coupling for extracting light scattered into asubstrate, in accordance with various teachings.

FIGS. 20A and 20B show embodiments for tiling frequency space with asubstrate (n=1.5) in one direction, in accordance with variousembodiments.

FIG. 21 show an exemplary tiling with almost complete frequency spacecoverage (NA=0.65, n=1.5), in accordance with present teachings.

FIG. 22 show another exemplary tiling with a larger NA objective lens(NA=0.95, n=1.5), in accordance with various embodiments.

FIG. 23 show another exemplary tiling strategy for high index substrate(n=3.6, collection NA=0.65, as in FIG. 18), in accordance with variousembodiments.

FIGS. 24A-24D show an example illumination and collectionconfigurations, where 24A shows an objective normal to theplane-parallel optical element surface, image frequencies up to(n_(pp)+NA)/λ can be captured, 248 shows an objective with tilt off fromthe optic axis, frequencies up to (n_(pp)+1)/λ, 24C shows an objectiveon the side of the plane-parallel optical element with grating,frequencies between (n_(pp)+1)/λ and 2n_(pp)/λ, and 24D shows spatialfrequency space coverage with regions collected with various geometriesindicated.

FIGS. 25A-O shows example arrangements for the plane-parallel opticalelement of FIG. 24A.

FIG. 26 shows an example geometry that shows access to collection highfrequencies propagating in the plane-parallel optical element thatcorrespond to small features.

FIG. 27 shows a plot depicting resolution restriction: normalized HPversus index of refraction for different NA (0.4, 0.8, 1.2), fixedplane-parallel optical element thickness: t=50 μm and field of view 32μm, where the solid lines represent the dependence described by thelower part of Eq. 13 and the dashed lines represent the dependencedescribed by the upper part of Eq. 13.

FIG. 28 shows a plot depicting resolution restriction: normalized HPversus index of refraction for different plane-parallel optical elementthickness: 10, 30, 100, 300 μm calculated with NA=0.4, F=32 μm indifferent synthetic aperture steps, where the long dashed linesrepresent the inside of synthetic aperture up to λ/[2(n_(pp)+3NA)], thedashed lines represent the inside of synthetic aperture up toλ/[2(n_(pp)+5NA)], and the dotted lines represent the inside ofsynthetic aperture up to λ/[2(n_(pp)+7NA)].

FIG. 29A shows an SEM image of periodic structure, HP=120 nm; 29B showsan IIM sub-image for t=2 mm and decoupling grating half-pitch of 280 nm.

FIG. 30A shows a model CD=120 nm structures and 308 shows an x-directionhigh frequency image.

FIGS. 31A-31D show difference in expansion of spectral package (120 nmfeatures) for different plane-parallel optical element thicknesses(n=1.5, F=64 μm), where for FIG. 31A, t=1 μm, image expansion˜3 timesand for FIG. 31B, t=5 μm, image expansion˜10 times; comparison offiltered image crosscuts (3002) with FIG. 31C showing crosscuts ofimages (3005) distorted by substrate propagation with t=1 μm, and FIG.31D showing crosscuts of images (3005) distorted by plane-paralleloptical element propagation with t=5 μm.

FIG. 32 show a plot depicting synthetic aperture guideline: normalizedsub-image bandwidth 2NA_(sub) versus normalized FOV for differentextracting gratings represented by center frequency

${{HP}_{c}\left( {g = \frac{n_{pp}{HP}_{c}}{\lambda}} \right)}.$

FIG. 33 shows FIGS. 33A-330 depicting restored images (CD=120 nm,n_(pp)=1.5), crosscuts and crosscut differences: FIG. 33A) t=1 μm, F=16μm—quality of the resorted image is good, FIG. 33B) t=5 μm, F=16 μm,quality of the resorted image is poor due to increased plane-paralleloptical element thickness; FIG. 33C) t=5 μm, F=32 μm, quality of theresorted image is improved as the result of increasing field of view.

FIG. 34 shows of plot of MSE versus HP of a 10-line pattern fordifferent plane-parallel optical element thicknesses, n=1.5; F=32 μm;λ=633 nm. 3% MSE considered as images with acceptable quality. 0.5 μmplane-parallel optical element allows restoration of images with 112 nmfeatures, 1 μm˜113.5 nm, 3 μm˜118 nm, 5 μm˜120 nm, 10 μm˜124 nm.

FIG. 35 shows a plot of HP versus n for different plane-parallel opticalelement thicknesses: 1, 5, 10 μm (F=32 μm), λ=633 nm. Plane-paralleloptical elements with higher n_(pp) allows resolution and restoration ofimages with smaller features.

FIG. 36 shows an example configuration for sectioning for 3D imagingwhere object A is in focus and object B is out of focus in accordancewith the present teachings.

FIG. 37 shows an example coordinate system for 3D imaging according tosome embodiments.

FIGS. 38A-E show example high frequency images imaged using theconfiguration of FIG. 36, where FIG. 38A shows image A in focus, FIG.38B shows image B defocused, FIG. 38C shows the sum of the two images:A+Bexp(iφ(λ₁)) C, FIG. 38D shows restored image A, and FIG. 38E showsrestored image B.

FIGS. 39A-F shows example pictures of recorded defocused high frequencyimage and electronically refocused one with corresponding models andcrosscuts, where FIG. 39A shows a defocused model, FIG. 39B shows adefocused experimental result, FIG. 39C shows crosscuts of defocusedmodel and experimental result, FIG. 39D shows a reconstructed model,FIG. 39E shows a reconstructed experimental result, and FIG. 39F showscrosscuts of reconstructed model and experimental results.

FIG. 40 shows an example multiple-axes, multiple-wavelengthconfiguration for 3D imaging in accordance with the present teachings.

FIG. 41 shows another configuration of 3D imaging incorporating asubstrate for mounting the object.

DESCRIPTION OF THE EMBODIMENTS

Reference will now be made in detail to the present embodiments,examples of which are illustrated in the accompanying drawings. Whereverpossible, the same reference numbers will be used throughout thedrawings to refer to the same or like parts.

Notwithstanding that the numerical ranges and parameters setting forththe broad scope of the invention are approximations, the numericalvalues set forth in the specific examples are reported as precisely aspossible. Any numerical value, however, inherently contains certainerrors necessarily resulting from the standard deviation found in theirrespective testing measurements. Moreover, all ranges disclosed hereinare to be understood to encompass any and all sub-ranges subsumedtherein. For example, a range of “less than 10” can include any and allsub-ranges between (and including) the minimum value of zero and themaximum value of 10, that is, any and all sub-ranges having a minimumvalue of equal to or greater than zero and a maximum value of equal toor less than 10, e.g., 1 to 5. In certain cases, the numerical values asstated for the parameter can take on negative values. In this case, theexample value of range stated as “less that 10” can assume negativevalues, e.g. −1, −2, −3, −10, −20, −30, etc.

FIG. 1 shows a prior art imaging interferometric microscopy (IIM)arrangement 100. As shown in FIG. 1, a collimated (equivalent tocoherent) illumination beam 110 is incident on an object 120 at an angleof incidence θ. In the illustrated case, θ is beyond the collectionangle of the objective lens 130 and an auxiliary optical system 135 isshown schematically to collect the zero-order transmission 109,appropriately adjust its divergence, direction, and phase and re-injectit onto the image plane 124 where it interferes with the diffractedbeams 101, 102, 103, 104 from the object 120 to construct a partialimage. Alternatively, instead of using the zero-order transmission 109,which might be blocked by the objective-lens 130 mount, a portion of theillumination beam 110 can be split off before the object 120 anddirected around the objective lens 130. The interference between thezero-order beam 109 and the diffracted beams 101, 102, 103, 104transmitted through the objective lens 130 can shift the collecteddiffracted information back to high frequency. As a result of thesquare-law intensity response, the resulting frequency coverage 140B canbe represented by a pair of circles 144, 146 of radius NA/λ shifted awayfrom zero frequency 142 by 2(2π)NA/λ as shown in FIG. 2B. FIG. 2B showsthe frequency space coverage 140B for off-axis coherent illumination,where frequencies beyond the lens bandpass are recorded in the sub-imageas a result of the interferometric reconstruction. For comparison, FIG.2A shows the frequency space coverage 140A for conventional normalincidence on-axis coherent illumination.

An object of the present teachings is to reduce or eliminate therequirement of the prior art for optical access to between the back ofthe objective lens and the image plane of the second optical system.This access is required for injecting the reference beam 109 in theprior art (FIG. 1). However, in many existing optical microscopes, thisregion is inaccessible. The structured illumination approach disclosedherein provides alternative methods of injecting the reference beam infront of the objective lens of the second optical system, therebysimplifying the application of imaging interferometric microscopy toexisting optical microscopy systems. Since both the diffracted beams andthe reference beams are not transmitted through the same objective,characterized by an NA, the high frequency image components arenecessarily shifted to lower frequency. This is similar to the use of anintermediate frequency in heterodyne radio receivers, but in the spatialfrequency rather than the temporal frequency domain. It is necessary toreset the spatial frequencies by signal processing after each sub-imageis captured in the electronic recording device. Additional advantages ofthis approach are that the pixel count requirements in the image planeare reduced, since only lower spatial frequencies, up to 2(2π)NA/λ, arerecorded, and the interferometer can be made smaller since all of thecomponents are on the front side of the objective lens, reducingvibrational effects on the interferometric reconstruction.

FIG. 3 shows an optical arrangement of the apparatus 200 for anexemplary structured illumination approach to IIM, according to variousembodiments of the present teachings. The apparatus 200 can include anobject 220 disposed on an object plane 222 on which a first surface of asubstrate 225 can be disposed, wherein the substrate 225 can becharacterized by a homogeneous refractive index (n_(sub)) and a surfacenormal 226. The apparatus 200 can also include a first optical systemincluding a source 211 and one or more optical components (not shown)disposed to provide a substantially coherent illumination 210 of theobject plane 222, the illumination 210 characterized by a wavelength kand a radius of curvature and disposed at one of a plurality of incidentwave vectors from about 0 to about 2π/λ with respect to a surface normalof the substrate and at a plurality of azimuth angles spanning 0 to 2π.Illumination 210 is diffracted by object 221 into a central,undiffracted 0^(th) order 209 and higher diffraction contributions whichcome from either higher orders of diffraction from a specific feature ofthe object or from additional spatial features of the object. Generally,smaller features give rise to larger diffraction angles. For convenienceof notation, these are collectively referred to as diffraction orders,including a 1^(st) order 201, a 2^(nd) order 202, a 3^(rd) order 203 anda 4^(th) order 204. The apparatus 200 can also include a second opticalsystem 230 disposed to collect portions of the illumination beamscattered from the object plane 222, the second optical system having anoptical axis 236 disposed at one of a plurality of center wave vectorsfrom about 0 to about 2π/λ with respect to the substrate surface normal226 and at the azimuth angle corresponding to the illumination of thefirst optical system, wherein the second optical system 230 ischaracterized by a numerical aperture (NA). In various embodiments, thesecond optical system 230 can include at least one objective lens. Theapparatus 200 can also include a third optical system represented bybeamsplitter 205 and mirror 206 disposed between the optical path of thefirst optical system and an entrance aperture of the second opticalsystem to provide interferometric reintroduction of a portion of thecoherent illumination (reference beam) 210′ into the second opticalsystem 230, wherein each of an amplitude, a phase, a radius ofcurvature, a path length and an angle of incidence of the reference beamcan be adjusted such that a correct reference beam can be present at aimage plane 224 of the second optical system 230. It is understood thatadditional optical components not shown are necessary to achieve thiscorrect reference beam. The reference beam 210′ is obtained by splittingoff a portion of the illumination beam 210 with beamsplitter 205 andredirecting the split-off beam with optical system 206 and the apparatus200 is configured so that the total path lengths of the illuminationbeam 210 and the reference beam 210′ from the beam splitter to the imageplane 224 are within the temporal coherence length of the source toinsure interferometric reconstruction of the sub-image. Illumination210′ is diffracted by substrate 225 into a central, undiffracted 0^(th)order 209′ and higher diffraction orders including a 1^(st) order 201′and a 2^(nd) order 202′. The apparatus can also include an electronicimage device 228 disposed at the image plane 224 of the second opticalsystem 230 that responds linearly to the local optical intensity andtransfers the local optical intensity map across the Image plane (asub-image) to a signal processor device 260 in electronic form. Invarious embodiments, the electronic image device 228 can be a chargedcoupled device (CCD) camera, a CMOS (complementary metal-oxidesemiconductor) camera, and any similar electronic focal plane arraydevice. The apparatus 200 can further include a device 250 for adjustingthe first, the second, and the third optical systems to collectsub-images for different pairs of the pluralities of incident (firstoptical system) and collection center (second optical system) wavevectors so as to sequentially obtain a plurality of sub-imagescorresponding to a plurality of regions of spatial frequency space. Invarious embodiments, the device can block/unblock various beams, rotatesubstrate etc. In some embodiments, the device can include mechanicalcomponents, such as, for example, motors. In other embodiments, thedevice can include electronic components, such as, for example, acousticmodulators or similar devices. The signal processor device 260 can alsobe arranged to sequentially receive the electronic form of thesub-images and manipulate the sub-images to correct for distortions andalterations introduced by the optical configuration, store, and combinethe plurality of sub-images corresponding to the plurality of regions ofspatial frequency space to create a composite image. In some otherembodiments, the signal processor device can include one or morecomputers. In some embodiments, the first, the second, and the thirdoptical systems can be arranged in a transmission configuration withrespect to the substrate surface. In other embodiments, the first, thesecond, and the third optical systems can be arranged in a reflectionconfiguration with respect to the substrate surface. Items 250, 235 and270 represent means to alter the various optical systems to correspondto different angles of incidence and scattering as described below.

In certain embodiments apparatus 200 for an exemplary structuredillumination approach to IIM can also include at least one knownreference object to cover a small part of the image field.

According to various embodiments, there is a method for structuralimaging interferometric microscopy. The method can include providing anobject 220 disposed over a planar substrate 225, wherein the substrate225 is characterized by a homogeneous refractive index (n) and a surfacenormal 226 and providing a first optical system to illuminate the object220 with substantially coherent illumination 210, the illuminationcharacterized by a wavelength λ and a radius of curvature and disposedat one of a plurality of incident wave vectors from about 0 to about2π/λ with respect to a surface normal of the substrate and at amultiplicity of azimuth angles spanning from about 0 to about 2π. Themethod can also include providing a second optical system 230 disposedto collect portions of the illumination scattered from the object plane222, the second optical system 230 having an optical axis 236 disposedat one of a plurality of center wave vectors from about 0 to about 2π/λwith respect to the substrate 225 surface normal 226 and at the azimuthangle corresponding to the illumination of the first optical system,wherein the second optical system 230 is disposed such that the object220 is substantially at the object plane 222 of the second opticalsystem 230 which is characterized by a numerical aperture (NA). Themethod can further include providing a third optical system disposedbetween the optical path of the first optical system and an entranceaperture of the second optical system to provide interferometricreintroduction of a portion of the coherent illumination (referencebeam) 210′ into the second optical system, wherein each of an amplitude,a phase, a radius of curvature and an angle of incidence of thereference can be adjusted such that a corrected reference wave ispresent at the image plane of the second optical system, wherein thecorrected reference beam 210′ and the illumination beam 210 are withinthe temporal coherence length of the source. The method can also includerecording a sub-image of the object 220 at an object plane 222 using anelectronic image device 228, wherein the sub-image is formed as a resultof interference between the scattering resulting from the coherentillumination of the object 220 and the reference beam 210′. The methodcan also include adjusting the first, the second, and the third opticalsystems to sequentially collect a plurality of sub-images correspondingto a plurality of regions of spatial frequency space, manipulating eachof the plurality of sub-images using a signal processor to correct fordistortions and alterations introduced by the optical configuration, andcombining the plurality of sub-images into a composite image to providea substantially faithful image of the object 220. In variousembodiments, the method can further include one or more processes ofsubtraction of dark field images, subtraction of background images,shifting of spatial frequencies in accordance with the opticalconfiguration, and elimination of one or more overlapping coverages ofthe frequency space wherein the elimination operations can be performedeither in the optical systems or in the signal processing. In someembodiments, the method can also include selecting the regions ofspatial frequency space to provide a more or less faithful image of theobject 220 in the object plane 222. One of ordinary skill in the artwould know that the regions of frequency space that are important varydepending on the object. For example for a Manhattan geometry pattern,there is less need to gather spectral information on the diagonals. See,for example, Neumann et al. in Optics Express, Vol. 16, No. 10, 2008 pp6785-6793 which describes a structural illumination for the extension ofimaging interferometric microscopy, the disclosure of which isincorporated by reference herein in its entirety.

FIG. 4 shows a flow diagram schematic, indicated generally by 400(a),(b) and (c), of structural illumination and restoration algorithms. Highspatial frequencies represented by diffracted beams from the off-axisillumination are mixed with the local oscillator beam, the dark field ofthe image is subtracted as is the low frequency image without its darkfield. Then the spatial frequencies are reset in Fourier space and thewhole image is reconstructed by combining high and low frequencysub-images.

To mathematically explain the structured illumination approach to IIM,first a simple mathematical description of a conventional coherentillumination microscopy image will be described and then themathematical description will be extended to the prior art IIMexperiment and finally to the structured illumination approach.

The total transmission through an arbitrary object (assumed to beperiodic on large scale to allow Fourier sums rather than Fourierintegrals) and illuminated by a plane wave at normal incidence can begiven by:

$\begin{matrix}{{\sum\limits_{{\forall k},{l \in k}}{A_{k,l}{\exp \left\lbrack {{\; {xk}\; \omega_{x}} + {\; {yl}\; \omega_{y}}} \right\rbrack}^{\; \gamma_{k,l}z}}} = {{A_{0,0}^{\; \gamma_{0,0}z}} + {\sum\limits_{k,{l \neq 0}}{A_{k,l}{\exp \left\lbrack {{\; {xk}\; \omega_{x}} + {\; {yl}\; \omega_{y}}} \right\rbrack}^{\; \gamma_{k,l}z}}}}} & (1)\end{matrix}$

where ω_(x), ω_(y) are the discrete spatial frequency increments of theFourier summation; x and y are orthogonal spatial coordinates;

$\gamma_{k,l} \equiv \sqrt{\left( \frac{2\pi \; n}{\lambda} \right)^{2} - \left( {k\; \omega_{x}} \right)^{2} - \left( {l\; \omega_{y}} \right)^{2}}$

with n the refractive index of the transmission medium (1 for air); R isthe set of integers, for which (|γ_(k,l)|)²>0, that is the range ofintegers for which the diffracted beams are within the bandpass of thetransmission medium and are propagating in the z-direction, away fromthe object. Note that this representation is a scalar approximation thatis appropriate as long as the angles do not get too large, and it isassumed below that all beams are polarized in the same direction. A morerigorous treatment is straightforward, but mathematically gets morecomplex and obscures the physical insight in these simpler equations.

The transmission through the optical system adds a filter factor:

$\begin{matrix}{{{T\left( {0;0} \right)}A_{0,0}^{\; \gamma_{0,0}z}} + {\sum\limits_{k,{l \neq 0}}{{T\left( {{k\; \omega_{x}};{l\; \omega_{y}}} \right)}A_{n,l}{\exp \left\lbrack {{\; {x\left( {k\; \omega_{x}} \right)}} + {\; l\; \omega_{y}y}} \right\rbrack}^{\; \gamma_{k,l}z}}}} & (2)\end{matrix}$

The transmission function of the objective lens is a simple bandpassfunction:

$\begin{matrix}\begin{matrix}{{T\left( {\omega_{X};\omega_{Y}} \right)} = {{1\mspace{14mu} {for}\mspace{14mu} \sqrt{\omega_{X}^{2} + \omega_{Y}^{2}}} \leq \omega_{{MA}\; X}}} \\{= \frac{2\pi \; {NA}}{\lambda}} \\{{= {0\mspace{14mu} {else}}},}\end{matrix} & (3)\end{matrix}$

and the final image intensity can be obtained by taking the squaremodulus of equation 2, viz:

$\begin{matrix}{{{I\left( {x,y,z} \right)} = {{\left\lbrack {T\left( {0,0} \right)} \right\rbrack^{2}{A_{0,0}}^{2}} + {\sum\limits_{k,l}{{T\left( {0,0} \right)}{T\left( {{k\; \omega_{x}},{l\; \omega_{y}}} \right)}A_{0,0}A_{k,l}^{*}{\exp \left\lbrack {- {\left( {{k\; \omega_{s}x} + {l\; \omega_{y}y}} \right)}} \right\rbrack}^{\text{?}{({\text{?} - \text{?}})}\text{?}}}} + {c.c.{+ {\sum\limits_{k,l}{\text{?}{T\left( {{k\; \omega_{x}},{l\; \omega_{y}}} \right)}{T\left( {{k^{\prime}\omega_{x}},{l^{\prime}\omega_{y}}} \right)}\text{?}A\text{?}\exp \left\{ {\left\lbrack {{\left( {k - k^{\prime}} \right)\omega_{x}x} + {\left( {l - l^{\prime}} \right)\omega_{y}y}} \right\rbrack} \right\} ^{{{({\text{?} - \text{?}})}}\text{?}}}}}} + {c.c}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (4)\end{matrix}$

Each of the three lines in this result has a simple physicalinterpretation. The top line is a constant independent of spatialcoordinates, equal to the average intensity of the pattern. This ensuresthat the intensity is always positive as physically required. The secondline represents the imaging terms that are retained. Finally the thirdline is the cross-correlation of the diffracted beams with themselvesequivalent to the dark field image that would be obtained if thezero-order diffraction (transmission) was blocked at the back pupilplane. The imaging terms are band-limited to transverse spatialfrequencies of (2π/λ)NA; the dark field terms extend out to (4π/λ)NA andare typically weaker in intensity than the imaging terms since for anobject with thickness <<λ, |A_(0,0)| is larger than any of thediffracted terms. In all of the summations the summation indices extendover all terms in R except for the zero-order term which has beenexplicitly separated out. Equation 4 gives the intensity over all spacebeyond the objective lens. The image is obtained in the back image plane(z=0) where the exponentials in γz vanish. The focusing information iscontained in these exponential terms and its characteristic length, thedepth-of-field, depends on the NA, as is well known. A Fourier opticsperspective provides additional insight into the three terms. The DCterm (top line) is a δ-function at the origin. The image terms fill acircle of radius 2πNA/λ as a result of the band-limited transmissionfunction. Finally, the dark-field image contains frequencies up to4πNA/λ as a result of the interference of the various diffracted orders.

It is well-known that additional, higher spatial frequency, informationcan be accessed with off-axis illumination. FIG. 1 shows a conventionalIIM arrangement 100, wherein a collimated illumination beam 110 can beincident on an object 120 at an angle of incidence θ. In particular inthe case of IIM, the offset angle is chosen such that the zero-ordertransmission (reflection) is beyond the lens (130)

${NA},{\omega_{offset} > {\frac{2\pi \; {NA}}{\lambda}.}}$

The result is that higher spatial frequency information is transmittedthrough the lens, but only a dark field image is recorded in atraditional coherent illumination microscopy configuration (without thereference beam 109). This is solved in IIM by introducing an auxiliaryoptical system 135, an interferometer that reinjects the zero-ordertransmission on the low-NA side of the lens to reset the spatialfrequencies. In practice it is simpler to reintroduce the zero-ordertransmission as an appropriately mode matched point source in the backpupil plane without actually using the transmitted beam which is oftenblocked by the objective lens mount. Effectively, the interferometerresults in a modified filter transfer function where the zero-order istransmitted even though it is outside the lens NA. The amplitude, thephase, and the offset position in the back focal plane of the objectivehave to be controlled to provide a correct sub-image. These are oftenset by using a nearby, known reference object along with the object ofinterest.

It is straightforward to extend the mathematical treatment to theoff-axis illumination case. Equation 2 can be modified to:

$\begin{matrix}{{{A_{0,0}^{\prime}^{{- \text{?}}\text{?}}^{{{(\text{?})}}s}} + {\text{?}{T\left( {{{k\; \omega_{x}} - \omega_{off}};{l\; \omega_{y}}} \right)}\text{?}{\exp \left\lbrack {{\; {x\left( {{k\; \omega_{z}} - \omega_{off}} \right)}} + {\; l\; \omega_{y}y}} \right\rbrack}^{\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (5)\end{matrix}$

where ω_(off)=2π sin(θ_(off))/λ is the frequency offset arising from theoff-axis illumination at angle θ_(off) (assumed in the x-direction), theprimes on the γs indicate that the propagation directions take intoaccount the offset illumination, and the prime on the A_(0,0) refers tothe re-injected 0-order.

Taking the square of equation 5 can provide the intensity on the imagingcamera:

$\begin{matrix}{\mspace{79mu} {{{A_{0,0}^{\prime}}^{2} + {\ldots \mspace{14mu} \left( {{dc}\mspace{14mu} {offset}} \right)}}{{\sum\limits_{k,{j \neq 0}}\; {A_{0,0}^{\prime}A_{k,l}^{*}{T\left( {{{k\; \omega_{x}} - \omega_{off}};{l\; \omega_{y}}} \right)}{\exp \left\lbrack {{\; k\; \omega_{x}} + {\; l\; \omega_{y}y}} \right\rbrack}\text{?}}} + {{c.c.{+ \ldots}}\mspace{14mu} ({imaging})}}{\sum\limits_{k,{j \neq 0}}\; {\text{?}A_{k,k}{T\left( {{{k\; \omega_{x}} - \omega_{off}};{l\; \omega_{y}}} \right)}A_{k,l}^{*}{T\left( {{{{k\;}^{\prime}\omega_{x}} - \omega_{off}};{{l\;}^{\prime}\omega_{y}}} \right)}{\exp \left\lbrack {{{\left( {k - k^{\prime}} \right)}\omega_{x}x} + {{\left( {l - l^{\prime}} \right)}\omega_{y}y}} \right\rbrack}\text{?}\mspace{14mu} \ldots \mspace{14mu} \left( {{dark}\mspace{14mu} {field}} \right)}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (6)\end{matrix}$

where the three terms on separate lines correspond to (top) a constantterm, (middle) the imaging terms and (bottom) the dark field image.Subtracting out the dark field terms (by taking an image with theinterferometer blocked so that only the third term survives) provides asub-image that accurately captures the spatial frequency components thatare transmitted through the optical system. Note that the imaging terms(middle line) are at the correct frequencies and that the offsetillumination angle has cancelled out of the expression except for thefilter transmission functions.

Changing both the illumination angle (and the angle of reintroduction)and the azimuthal angle changes the offset allowing recording of adifferent region of frequency space. Specifically, for Manhattangeometry (x,y oriented patterns) a second offset exposure to capture thehigh spatial frequencies in the y-direction, that is with the substraterotated by π/2, can be used. Additional spatial frequency terms can becaptured with large illumination angles.

Referring back to the FIG. 3, in the exemplary structured illuminationapproach to IIM, there can be two coherent illumination beams 210, 210′,the first beam 210 can be at the same offset as in the previous exampleso that ω_(offset) is >NA/λ, and the second beam 210′ can be at themaximum offset that fits through the lens ω_(off)≦NA/λ, denoted asω_(NA) in the equation. Then the fields are:

$\begin{matrix}{{{A_{0,0}{\exp \left( {{- {\omega}_{off}}x} \right)}\text{?}} + {\sum\limits_{k,{l \neq 0}}\; {A_{k,l}{\exp \left\lbrack {{{\left( {{k\; \omega_{x}} - \omega_{off}} \right)}x} + {\; l\; \omega_{y}y}} \right\rbrack}\text{?}}} + {B_{0,0}{\exp \left( {{- {\omega}_{NA}}x} \right)}\text{?}} + {\sum\limits_{p,{r \neq 0}}\; {B_{p,r}{\exp \left\lbrack {{{\left( {{p\; \omega_{x}} - \omega_{NA}} \right)}x} + {\; r\; \omega_{y}y}} \right\rbrack}\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (7)\end{matrix}$

where the series with coefficients A_(k,j) are due to the first offsetbeam (210) and the second series with the coefficients B_(p,q) are dueto the second offset beam (210′) and squaring while taking advantage ofthe fact that without the interferometer the A_(0,0) beam 209 is nottransmitted to the objective image plane while the B_(0,0) beam 209′ istransmitted through the lens 230 gives:

$\begin{matrix}{\mspace{79mu} {{\lbrack I\rbrack \mspace{14mu} \begin{Bmatrix}{{B_{0,0}}^{2} + {\sum\limits_{p,{r \neq 0}}\; {B_{0,0}B_{p,r}^{*}{T\left( {{{p\; \omega_{x}} - \omega_{NA}};{r\; \omega_{y}}} \right)}}}} \\{{{\exp \left\lbrack {\left( {{p\; \omega_{x}} + {r\; \omega_{y}y}} \right)} \right\rbrack}\text{?}} + {c.c. +}} \\{\sum\limits_{p,{r \neq 0}}\; {\sum\limits_{p^{\prime},{r^{\prime} \neq 0}}\; {B_{p,y}B_{p,r}^{*}{T\left( {{{p\; \omega_{s}} - \omega_{NA}};{r\; \omega_{y}}} \right)}}}} \\{T\left( {{{p^{\prime}\omega_{x}} - \omega_{NA}};{r^{\prime}\omega_{y}}} \right)} \\{{\exp \left\lbrack {{{\left( {p - p^{\prime}} \right)}x} + {{\left( {r - r^{\prime}} \right)}y}} \right\rbrack}\text{?}}\end{Bmatrix}} + {\lbrack{II}\rbrack \mspace{14mu} \left\{ {{\sum\limits_{k,l}\; {B_{0,0}A_{k,l}^{*}{T\left( {{{l\; \omega_{x}} - \omega_{off}};{n\; \omega_{y}}} \right)}{\exp \left\lbrack {{{- {\left( {{k\; \omega_{x}} - \omega_{off} + \omega_{NA}} \right)}}x} - {\; k\; \omega_{y}y}} \right\rbrack}\text{?}}} + {c.c}} \right\}} + {({III})\mspace{14mu} {\sum\limits_{k,l}\; {\sum\limits_{k^{\prime},l^{\prime}}\; {A_{k,l}A_{k^{\prime},l^{\prime}}^{*}{T\left( {{{k\; \omega_{x}} - \omega_{off}};{l\; \omega_{y}}} \right)}{T\left( {{{k^{\prime}\omega_{x}} - \omega_{off}};{l^{\prime}\omega_{y}}} \right)}{\exp \left\lbrack {{{\left( {k - k^{\prime}} \right)}\omega_{x}x} + {{\left( {l - l^{\prime}} \right)}\omega_{y}y}} \right\rbrack}\text{?}}}}} + {{c.c.({IV})}\mspace{14mu} {\sum\limits_{k,l}\; {\sum\limits_{p,{r \neq 0}}\; {A_{k,l}B_{p,r}^{*}{T\left( {{{k\; \omega_{x}} - \omega_{off}};{l\; \omega_{y}}} \right)}{T\left( {{{p\; \omega_{x}} - \omega_{NA}};{r\; \omega_{y}}} \right)} \times {\exp \left\lbrack {{{\left( {k - p} \right)}\omega_{x}x\; {\left( {\omega_{NA} - \omega_{off}} \right)}x} + {{\left( {l - r} \right)}\omega_{x}}} \right\rbrack}\text{?}}}}} + {{c.c.\text{?}}\text{indicates text missing or illegible when filed}}}} & (8)\end{matrix}$

The first three terms in the upper bracket, labeled [I], in equation 8are the result of the off-axis illumination at the edge of the pupil.This image can be measured independently by blocking the extreme offaxis beam and subtracted from the result. The term labeled [II] is thedesired information, the image terms beating against a zero-order beam;because the zero-order beam is not at the correct angle to reset thefrequencies to match the object frequencies (adjusted for magnification)there is a shift between the observed and the actual image planefrequencies {exp[i(ω_(NA)−ω_(off))x]} that will need to be fixedcomputationally (e.g. one is measuring the Fourier components at anintermediate frequency as detailed above). [III] is the dark field fromthe extreme off-axis illumination. Finally the last term, [IV] is thecross-correlated dark field from the two illumination beams.

To remove the unwanted terms in equation 8, five strategies can be used.However, these are not intended to be all-inclusive and otherpossibilities may exist. These are illustrated schematically in FIGS.5A-5E. There are two general approaches, in the FIGS. 5A-5C, thereference beam is added before the object plane. This adds to someadditional complexity in that the off axis and the reference beams giverise to diffracted information and it is necessary to separate out theinformation corresponding to the diffraction from the reference beamfrom the off axis beam. This can be accomplished as shown in the schemeoutlined in FIG. 4. In FIGS. 5D and 5E, the reference beam is addedafter the object plane and before the entrance to the collection lens.In these configurations, the reference beam does not illuminate theobject and hence there is no additional diffraction. This simplifies theanalysis, but at the cost of adding additional optical components in theregion of limited access.

FIG. 5A shows the first embodiment, wherein the third optical system500A can further include a first beamsplitter disposed in the opticalpath of the first optical system to collect a portion of the coherentillumination, one or more optical components to direct the portion ofthe coherent illumination as a reference beam 510′ to illuminate theobject 520 at an angle θ corresponding to less than the entrance angularaperture (<˜sin⁻¹ NA) of the second optical system 530, and a dynamic(adjustable on/off) physical block 550 disposed in a back pupil plane ofthe second optical system 530 to alternately block and unblock a smallportion of the pupil aperture corresponding to the position of thereference beam 510 in the aperture. One of the advantages of thisembodiment is that all of the information can be retained. However, thisembodiment requires access to the illumination system pupil, in the caseshown in FIG. 5A, the objective pupil has been relayed to an auxiliaryplane where it might be easier to access. The details of this opticalconfiguration will depend on the optical construction of the objectivelens.

FIG. 5B shows the second embodiment 500B, wherein both illuminationbeams can be shifted slightly using a prism 560 so that the zero order209′ can be blocked but there is no change in the exponential factor,only in the transmission factors. Using the first and secondembodiments, one can obtain and subtract dark field optical intensitiesfrom the image formed by interference of low and high frequencies (thesecond, fourth, and fifth terms of equation 6). Then one can subtractlow frequency image without dark field and restore high frequency imageby shifting frequencies in Fourier space. The second embodiment can beimplemented easily and does not require any access to the objectivepupil plane but it has some image-dependent information loss associatewith the shifting of the illumination angles. As shown in FIG. 5B, theprism is located in between the object 520 and the entrance aperture ofthe objective lens 530; alternatively it can be located before theobject 520. Depending on the specifics of the object 520, it may beadvantageous to dither the position of only the reference zero-orderbeam or of both zero-order beams.

FIG. 5C shows yet another embodiment using a guided-mode filter(k-vector filter) 582 to block the zero order transmission just beforethe objective 530 and transmit the diffracted information at all otherangles. FIG. 6 shows an exemplary experimental un-optimized k-vectorfilter characteristic of a silicon-nitride-on-glass guided moderesonance filter, with a narrow angular width of the coupling. U.S. Pat.No. 5,216,680 discloses guided mode filter which can be used as anoptical filter with very narrow line width and as an efficient opticalswitch, the disclosure of which is incorporated by reference herein inits entirety. Referring back to FIG. 5C, it is possible to switch thezero-order on and off by mechanical dithering of the angular position orby dithering by a small degree of rotation around the optical axis,shown generally by 590. This will allow identification of the source ofthe diffracted waves in the sub-image. Accordingly, the exemplary thirdoptical system 500C of the apparatus 200 in accordance with variousembodiments, can further include one or more optical components todirect the portion of the coherent illumination as a reference beam toilluminate the object 520 at an angle θ less than the entrance angularaperture (<˜sin⁻¹ NA) of said second collection optical system 530, theguided-mode resonance filter (k-vector filter) 582 disposed between theobject plane 522 and a collection lens of the second optical system 530,and another device 527 to adjust the position, tilt and rotation of theguided-mode resonance filter 582 between positions, tilts and rotationsin which it alternately transmits and blocks the portion of thereference beam transmitted through the object plane.

FIG. 5D shows yet another exemplary third optical system 500D of theapparatus 200 in accordance with various embodiments. The third opticalsystem 5000 can further include a first beamsplitter disposed in theoptical path of the first optical system to collect a portion of thecoherent illumination, one or more transfer optics disposed between thefirst optical system and the second optical system, and at least one ofa grating 584 or a grating on a waveguide disposed between the objectplane 522 and a front aperture of the collection lens (objective) of thesecond optical system 530 to reintroduce the portion of the coherentillumination as a reference beam into the second optical system 530 atan angle θ less than the entrance angular aperture (<˜sin⁻¹ NA) of thesecond optical system. In various embodiments, the grating 584 can havea short pitch (high spatial frequency) to avoid diffraction of the wavesincident onto the grating 584 into new directions that are captured bythe objective lens of the second optical system 530. A major advantageof this method is that it does not require switchable gratings ormechanical actuation of the filter, since modulation is by simpleblocking of the incident beam.

FIG. 5E shows the fifth embodiment, wherein the zero order 510′ can bere-injected after the object and just in front of the objective by abeamsplitter 570. Accordingly, the third optical system can include asecond beamsplitter 570 disposed between the object plane 522 and afront aperture of a collection lens (objective) of the second opticalsystem 530 to reintroduce the portion of the coherent illumination as areference beam 510′ into the second optical system 530 at an angle θless than the entrance angular aperture (<˜sin⁻¹ NA) of the secondoptical system 530. The beamsplitter 570 can eliminate all of thediffracted beams associated with the local oscillator, B^(p,r)=0,∀p,r≠0, and simplifies equation 8. The third embodiment does not containthe first, second and fifth terms at all, so it is the most robust forthe image processing, but images can be distorted by aberration causedby the beamsplitter; this aberration can be corrected with additionaloptical components. Using a very thin beamsplitter, e.g., a pellicle,eliminates much of the aberration associated with an expanding beampassing through a tilted thin plate. The use of a beamsplitter impactsthe working distance of the objective, but the depth-of-field andfield-of-view advantages of IIM are retained.

FIGS. 7A and 7B show the frequency space coverage for the structuredillumination approach to IIM shown in FIG. 3 and using the firstembodiment as shown in FIG. 5A. All of the recorded frequencies arewithin the bandpass of the objective lens. The two offset circles 741,742 in FIG. 7A correspond to coverage 740A of both the intermediatefrequency imaging terms and the offset frequency imaging terms beatingwith the intermediate frequency local oscillator. FIG. 7B shows thecoverage 7408 after the unwanted dark field and local oscillatorself-imaging terms have been removed and the spatial frequencies havebeen reset to their correct values.

FIGS. 8A and 8B illustrate the frequency space coverage of partialimages. FIG. 8A shows an illustration of a Manhattan (x, y) geometrypattern 800A used for image resolution exploration consisting of fivenested “ells” and a large box. The lines and spaces of the “ells” areabout 240 nm. FIG. 8B, indicated generally at 800B, shows the intensityFourier space components of the pattern 800A, mapped onto the frequencyspace coverage of the imaging system using a NA=0.4 objective and anillumination wavelength of 633 nm (HeNe laser source). The resolutionlimit of this microscope system with conventional illumination is˜0.6λ/NA (˜950 nm). The two circles at radii of NA/λ (0.4/λ) and 2NA/λ(0.8/λ) correspond to the approximate frequency space limits forcoherent and incoherent illumination, respectively, and reflect thelow-pass transmission characteristic of the objective lens. The innersets of small shifted circles (radius NA/λ) in FIG. 8B, that extend from0.3NA/λ to 3NA/λ (±1.2/λ) in the x- and y-directions, show the frequencyspace coverage added with two offset partial images, one in eachdirection. The imaging is single side-band, only the diffracted planewaves to one side of the object are collected (opposite to the tilt ofthe illumination beam), the square law (intensity) response of the imageformation and detection process restores the conjugate frequency spacecomponents, resulting in the two symmetrically displaced circles in FIG.8B for each partial image. The offset (off-axis tilt) for these imageswas chosen at 2(2π)NA/λ to ensure that there was no overlap between thespectral coverage of the low-frequency partial image (extending out toNA/λ) and the offset images. As discussed previously, improved imagescan be obtained by subtracting the dark-field components of the image(with the zero-order transmission blocked). In the present embodiments,this provided a cosmetic, not a dramatic, improvement to the images.Additional frequency space coverage is available with a second pair ofoff-axis images, represented by the outer sets of shifted circles, witha larger tilt of the illumination plane wave, approaching grazingincidence (limited to 80° by practical considerations such as Fresnelreflectivity in the present experiment). The maximum frequency coveragein these images extends to [sin(80)+NA]/λ=(0.98+NA)/λ=(1.38/λ). Thefrequency-space coverage of the outer circles may be necessary tocapture the fundamental frequency components of the line-space portionof this pattern. There is significant overlap between the frequencycoverage of the first and second set of partial images as illustrated inFIG. 8B. To provide a faithful image, the double coverage of frequencyspace associated with the image spectral overlaps can be excluded. Thiscan be accomplished by filtering the images either optically (withappropriate apertures in the back focal plane) or electronically oncethe images are recorded. Importantly, since each of the partial imagesinvolves only the NA of the objective, this imaging concept retains theworking distance, depth-of-field and field-of-view associated with thelow-NA objective, but has a resolution beyond that achievable with eventhe highest NA objective using traditional illumination approaches.

FIGS. 9A-9F show the preliminary results of an experiment using anNA=0.4 objective with a He—Ne laser illumination (λ=633 nm) and withabout a 240 nm critical dimension structure with correspondingsimulations using the configuration presented in FIG. 50, blocking thezero-order beam of the reference in the objective lens pupil. FIG. 9A isthe mixed image corresponding to the interference of the low and highimages and FIG. 9B is the corresponding simulation result. FIG. 9C isthe image after subtraction dark field and low frequency image and FIG.9D is the corresponding simulation result. FIG. 9E is restored highfrequency image and FIG. OF is the corresponding simulated result.

Similarly, results using dynamic (adjustable on/off) physical blockpresented in FIG. 5A are shown in FIG. 10A. The same 260- and 240-nmobjects are imaged as in FIG. 9C; only the final results after the darkfield subtraction, frequency shifting correction and sub-imagecombination are shown. The corresponding cross-cuts are shown in FIG.10B. A total of four offset images, two each in the x- and y-directions,with θ_(ill)=53° and 80° were used along with a 0.4 NA objective. Asdiscussed previously, this configuration provided resolution to <˜240 nmCD. There is overlap in the frequency space coverage between these twoexposures and electronic frequency space filtering is used to assure auniform coverage of frequency space. The present Manhattan geometrystructure has spectral content concentrated along the x- andy-directions, so the offset illuminations were restricted to thosedirections. Adding additional frequency-space coverage for arbitrarilyshaped structures can be accomplished by taking additional sub-imageswith rotation of the object in the (x,y) plane. The spatial frequencycontent of the image covers a wide range as a result of the large box(at 10× the linewidth of the linespace structures). The reconstructedimage of the same structures obtained by the method with thebeamsplitter configuration presented in FIG. 5E is shown in FIG. 11A anda crosscut of the image with corresponding simulation is shown in FIG.11B. The quality of the results for both methods is quite comparable.The second method retains a long working distance, but requires accessto the Imaging system pupil for blocking the zero-order. The firstmethod does not require any modification to the traditional microscopycomponents, but has reduced working distance due to the beamsplitter infront of the objective. There are some extra features experimentally ascompared to the model due to the lack of precision in mutual phasedetermination between the sub-images and speckle effects from thecoherent illumination. These issues can be reduced by using improvedarrangements and lower coherence sources. There are other possiblealternatives; the optimum choice will depend on the specifics of theobject and the constraints of specific optical systems.

The embodiments discussed so far provide spatial frequency coverage upto 2π(sin(θ_(ill))+NA)/λ≦2π(1+NA)/λ; that is the maximum illuminationangle offset can be set close to 90° (providing the “1”) and the maximumangle collected by the objective lens corresponds to sin⁻¹(NA). As waspreviously disclosed in relation to the interferometric implementationof IIM, additional spatial frequencies are available by tilting theobject plane relative to the objective lens axis. This allows collectionof spatial frequencies up to 4π/λ, independent of the lens NA. The costis a more complex signal processing requirement since the tilted objectplane results in a nonlinear mapping of spatial frequencies from theobject plane to the laboratory image that must be corrected to achieve agood image. This mapping has been discussed previously. The additionalfrequency space (and hence smaller image features) are available in thestructured illumination embodiments of IIM disclosed herein.

Immersion microscopy is well known to provide higher spatial frequenciesby a factor of the refractive index of the immersion medium, n, therebyextending the spatial frequency range to as high as 2 n/λ. Again theadvantages of immersion are directly applicable to structuredillumination IIM.

Traditionally immersion microscopy has been practiced in reflection witha liquid medium on top of the object, or in transmission where advantageis taken of the high refractive index of the substrate (n_(sub)) as wellas that of a liquid on top of the object. An intermediate possibility isto use the refractive index of the substrate without an immersion fluid.In this case the spatial frequency range is extended to2π(n_(sub)+NA)/λ.

FIG. 12 shows an exemplary apparatus 1200 for microscopy with an IIMarrangement with illumination by evanescent waves extending from asubstrate, where 12A and B refers to alternate positions of thecollection lens, according to various embodiments of the presentteachings. The apparatus 1200 can include an object plane 1222 on whichcan be disposed a first surface of a substrate 1225, wherein thesubstrate 1225, upon which is an object 1220, is characterized by ahomogeneous refractive index (n_(sub)) and a surface normal 1226. Theapparatus 1200 can also include a first optical system, including prisms1262 and substrate 1265, disposed to provide a substantially coherentevanescent wave illumination of the object 1220, the illumination 1210characterized by a wavelength λ and a radius of curvature and disposedat one of a plurality of incident wave vectors from about 2π/λ to about2πn_(sub)/λ with respect to a surface normal of the substrate and at aplurality of azimuth angles spanning from about 0 to about 2π, whereinthe plurality of incident wave vectors correspond to angles beyond atotal internal reflection angle θ_(c) of the substrate. The apparatus1200 can also include a second optical system 1230 disposed to collectportions of the illumination 1208 scattered from the object plane 1222,the second optical system 1230 having an optical axis 1236 disposed atone of a plurality of center wave vectors from about 0 to about 2π/λ (orangles from about 0 to about π) with respect to the substrate 1225surface normal and at the azimuth angle corresponding to theillumination of the first optical system, wherein the second opticalsystem 1230 is characterized by a numerical aperture (NA). FIG. 12Bshows arrangement with tilted optical axis 1236′. The apparatus 1200 canalso include a third optical system disposed in an optical path of thefirst optical system to provide interferometric reintroduction of aportion of the coherent illumination (reference beam) into the secondoptical system 1230 or 1230′, wherein each of an amplitude, a phase, aradius of curvature and an angle of incidence of the reference isadjusted such that a correct reference wave is present at the imageplane of the second optical system. The apparatus 1200 can furtherinclude an electronic image device disposed at an image plane of thesecond optical system that responds linearly to the local opticalintensity and transfers the local optical intensity map across the imageplane (a sub-image) to a signal processor device in electronic form. Theapparatus 1200 can also include a device for adjusting the first, thesecond, and the third optical systems to collect sub-images fordifferent pairs of the pluralities of incident (first optical system)and collection center (second optical system) wave vectors so as tosequentially obtain a plurality of sub-images corresponding to aplurality of regions of spatial frequency space and an electronic deviceto sequentially receive the electronic form of the sub-images andmanipulate the sub-images to correct for distortions and alterationsintroduced by the optical configuration, store, and combine theplurality of sub-images corresponding to the plurality of regions ofspatial frequency space to create a composite image corresponding to asynthetic aperture that is larger than the physical aperture of thecollection lens 1230 or 1230′.

In some embodiments, the third optical system can further include afirst beamsplitter disposed in the optical path of the first opticalsystem before the object to collect a portion of the coherentillumination and one or more optics disposed between the first opticalsystem and the second optical system 1230 are prisms 1262 within firstoptical system used to inject into substrate at angles beyond totalinternal reflection to interferometrically reintroduce the portion ofthe coherent illumination as a reference beam into the second opticalsystem 1230 in a position after the exit aperture of a collection(objective) lens, wherein the reintroduction is at one of a positioncorresponding to a position a zero-order beam would have had if it hadbeen transmitted through a higher NA lens of the second optical system1230 or an aliased position to reduce pixel requirements of theelectronic image device, wherein the signal processor is adjusted tocompensate for this spatial frequency aliasing (the same concept as thelocal oscillator frequency introduced earlier). In other embodiments,the third optical system of the apparatus 1200 can include one of theconfigurations shown in FIGS. 5A-5E.

In certain embodiments apparatus 1200 for microscopy with an IIMarrangement with illumination by evanescent waves extending from asubstrate can also include at least one known reference object to covera small part of the image field.

According to various embodiments, there is a method for microscopy byevanescent illumination through a substrate. The method can includeproviding an object 1220 disposed on a surface of a planar substrate1225 characterized by a homogeneous refractive index (n_(sub)) and asurface normal 1226 and providing a first optical system disposed toprovide an evanescent wave illumination of the object plane 1222 byproviding a substantially coherent illumination of the object plane1222, the Illumination characterized by a wavelength λ and a radius ofcurvature and disposed at one of a plurality of incident wave vectorsfrom about 2π/λ about 2πn_(sub)/λ with respect to a surface normal ofthe substrate and at a multiplicity of azimuth angles spanning 0 to 2π,wherein the plurality of incident wave vectors correspond to anglesbeyond a total internal reflection angle θ_(c) of the substrate. Themethod can further include providing a second optical system 1230 havingan optical axis 1236 disposed at one of a plurality of center wavevectors from about 0 to about 2π/λ with respect to the normal to theplane parallel optical element, wherein the second optical system 1230is characterized by a numerical aperture (NA). The method can alsoinclude providing a third optical system disposed in an optical path ofthe first optical system to provide interferometric reintroduction of aportion of the coherent plane wave illumination (reference beam) intothe second optical system 1230, wherein the amplitude, phase, andposition of the reintroduced illumination wave in the image plane of thesecond optical system 1230 can be adjusted. The method can furtherinclude recording a sub-image of the object 1220 at an object plane 1222using an electronic image device, wherein the sub-image is formed as aresult of interference of the scattering from the coherent plane waveillumination of the object and the reference beam; adjusting the first,the second, and the third optical systems to sequentially collect aplurality of sub-images corresponding to a plurality of regions ofspatial frequency space; manipulating each of the plurality ofsub-images using a signal processor to correct for distortions andalterations introduced by the optical configuration; and combining theplurality of sub-images into a composite image to provide asubstantially faithful image of the object. In various embodiments, themethod can further include one or more processes of subtraction of darkfield images, subtraction of background images, shifting of spatialfrequencies in accordance with the optical configuration, andelimination of one or more overlapping coverages of the frequency spacewherein the elimination operations can be performed either in theoptical systems or in the signal processing. In some embodiments, themethod can also include selection of the regions of spatial frequencyspace to provide a more or less faithful image of the object in theobject plane. Neumann et al. in Optics Express, Vol. 16, No. 25, 2008 pp20477-20483 describes an evanescent wave illumination for furtherextending the resolution limit of imaging interferometric microscopy toλ/2(n_(sub)+1), the disclosure of which is incorporated herein byreference in its entirety.

In various embodiments, the step of providing an object 1220 disposed ona surface of a planar substrate 1225 can include providing a claddinglayer surrounding the object 1220 and the object 1220 disposed over thesubstrate 1225. The extent of excitation region due to evanescent waveillumination, normal to the interface is given by an exponential decayfunction with a 1/e length of λ/2π√{square root over (n_(pp) ² sin²θ−n_(clad) ²)}, where n_(sub) is the refractive index of the substrateand n_(clad) is the refractive index of the superstrate or claddingmaterial surrounding the object 1220. The spatial localization canprovide benefit, for example in TIRF (total internal reflectionfluorescence) the localization is much larger than can be achieved witha simple focus or even with confocal microscopy. In other cases, thisdecay length can be a restriction, for example, in lithography studieswhere there might be multiple layers of material (bottom AR coating andphotoresist for example) and the structural variation between theselayers is of interest. Hence, the addition of a cladding layersurrounding the object can allow some degree of tuning of the decaylength, and thereby control the signal to noise ratio.

FIGS. 13A-13C shows several exemplary techniques for part of the firstoptical system to provide illumination through the substrate 1325. FIG.13A shows coupling of incident beam 1310 through a side 1326 of thesubstrate 1325, which can be polished at an angle different from normalto the object 1320; in other words the substrate 1325 can be a prism.FIG. 13B shows one or more gratings 1364 on a side of the substrate 1325the same as that where the object 1320 can be located. Alternatively,the gratings 1364 can be placed on a side opposite to that of the object1320. FIG. 13C shows coupling of the incident beam 1310 using one ormore prisms 1362.

FIG. 14A shows a Manhattan (x-, y-geometry) test pattern, scaled todifferent dimensions. The Fourier intensity transform of this patternfor a linewidth (critical dimension or CD) of 180 nm is shown in FIG.14B and for a CD of 150 nm in FIG. 13C. The circles in FIGS. 14B and 14Ccorrespond to the bandpass limits of various microscopy configurations.The circle in the center of FIG. 14B, with a radius of NA/A=0.4/A,corresponds to the Abbé-limit spatial frequency range captured withon-axis coherent illumination (NA_(ill)=0). The inner set of shiftedcircles in FIG. 14B (only single sidebands are shown for clarity; thecomplex conjugate regions are covered as well) correspond to IIM withoff-axis illumination beams at θ_(ill)=53° in the x, y-directions thatextend the frequency coverage to a radius 3NA/λ˜1.2/λ. Additionalfrequency space coverage (second pair of circles) is available usingevanescent wave illumination extending the frequency space coverage to aradius of (n_(sub) sin θ_(ill)+NA)/λ˜1.87/λ (with θ_(ill)=76°) withouttilt of the microscope optical axis. The frequency space coverage alongwith the corresponding structure Fourier intensity plot for thestructure with CD=150 nm is shown in FIG. 14C. The third pair ofoff-axis sub-images in FIG. 14C correspond to the tilted optical axis.This frequency region is elliptical rather than circular, due tononparaxial and conical diffraction effects associated with the off-axisoptical system.

FIG. 15A shows the experimental result for an object containing both180- and 170-nm CD structures in a single large-field image using theapparatus of FIG. 12A (two pairs of offset illumination, one at 53° inair and one at 76° in the substrate and collection with the optical axisalong the substrate's surface normal as shown in FIG. 12A. The 180-nm CDobject is within the bandwidth capabilities of this optical system whilethe 170-nm CD object has significant spatial frequencies that extendbeyond the optical system bandwidth and so is not fully resolved. Thefive nested “ell” shapes are distinguishable for the 180-nm CD, but notfor the 170-nm CD. The positions of the two objects are correctlyrestored by the image restoration procedure as is evident from the goodpositional overlap between the experimental and theoretical cross-cutsin FIG. 15B.

FIG. 16A shows reconstructed high frequency image of a 150 nm structureusing evanescent illumination and a tilted optical system, shown in FIG.12B, with the highest spatial frequencies collected with the opticalaxis tilted with respect to the substrate's surface normal. FIG. 16Bshows high frequency image simulation of a 150 nm structure usingevanescent illumination and a tilted optical system, shown in FIG. 12B.FIG. 16C shows experimental and simulation cross-cuts of images shown inFIGS. 16A and 168. FIG. 16D shows reconstructed composite image of a 150nm structure using evanescent illumination and a tilted optical system,shown in FIG. 12B. FIG. 16E shows composite image simulation of a 150 nmstructure using evanescent illumination and a tilted optical system,shown in FIG. 12B. FIG. 16F shows experimental and simulation cross-cutsof images shown in FIGS. 16D and 16E.

Evanescent illumination can be combined with structural illuminationeliminating the need for access to the back focal plane. This moves theinterferometer to the front of the objective lens and makes IIM readilyadaptable to existing microscopes. Structural illumination is roughlyequivalent to recording the spectral information at an intermediatefrequency; additional computation is required to reset the frequencies.But this frequency shifting can reduce the camera pixel size and countrequirements. Evanescent wave illumination can be used to extend theresolution of IIM to λ/2(n+1). Furthermore, IIM provides an importantadvantage over conventional immersion microscopy techniques. Since onlya relatively small region of frequency space (˜NA/λ) is recorded in eachsub-image, the aberration requirements on the objective lens aredramatically reduced. Hence, a simple set of prisms or gratings can beused to extract, and conventional air-based lenses to capture, theinformation. As is always the case, there is a trade-off between thenumber of sub-images and the NA of the objective lens.

FIG. 17 shows the possible increase of NA_(eff), drawn for a 0.4 NAsystem. As the frequency coverage is extended, the use of higher NAlenses can reduce the number of sub-images required for a more completecoverage of frequency space. Of course the required coverage isdependent on the pattern, and there are some applications, for examplein metrology for integrated circuits, where coverage of a subset of thefull frequency space is appropriate, where the range of spatialfrequencies in the object are limited by lithographic consideration.

There are diffracted beams corresponding to even larger spatialfrequencies (smaller features) scattered back into the substrate atangles larger than the critical angle. These beams are totallyinternally reflected and are not accessible. FIG. 18 shows anotherexemplary IIM optical arrangement for an apparatus 1800 for microscopythat provides access to the higher spatial frequency terms and therebyprovides higher resolution, according to various embodiments of thepresent teachings. The apparatus 1800 can include an object plane 1822on which can be disposed a first surface of a planar substrate 1825,wherein the substrate 1825 is characterized by a homogeneous refractiveindex (n_(sub)) and a normal 1826. The apparatus 1800 can also include afirst optical system disposed to provide a substantially coherentillumination of the object plane, the illumination characterized by awavelength λ and a radius of curvature and disposed at one of aplurality of incident wave vectors from about 0 to about 2πn_(sub)/λwith respect to a surface normal 1826 of the substrate 1825 and at aplurality of azimuth angles spanning from about 0 to about 2π. Theapparatus 1800 can further include at least one grating 1864 on the sideof the substrate 1825 opposite the object plane 1822, wherein eachgrating 1864 is characterized by a period, a depth, a grating profile, aposition, and an extent to further transform the scattered waves in thesubstrate reflected from the illumination by the object into propagatingwaves in the medium below the substrate. In some embodiments, the mediumbelow the substrate 1825 can be air. In other embodiments, the mediumcan be a vacuum. However, the medium can include any other suitablematerial. The apparatus 1800 can further include a second optical system1830 having an optical axis 1836 disposed at one of a plurality ofcenter wave vectors from about 0 to about 2π/λ (or angles from about 0to about π) with respect to the surface normal 1826, wherein the secondoptical system 1830 can include one or more gratings 1864 on the secondside of the substrate 1825 and is characterized by a numerical aperture(NA). The apparatus 1800 can also include a third optical systemdisposed in an optical path of the first optical system to provideinterferometric reintroduction of a portion of the coherent illumination(reference beam) into the second optical system 1830, wherein each of anamplitude, a phase, a radius of curvature, path length, and an angle ofincidence of the reference can be adjusted such that a correct referencewave is present at the image plane of the second optical system. Theapparatus 1800 can further include an electronic image device disposedat an image plane of the second optical system 1830 that respondslinearly to the local optical intensity and transfers the local opticalintensity map across the Image plane (a sub-image) to a signal processordevice in electronic form, a signal processor that receives theelectronic form of the sub-image and manipulates the sub-image tocorrect for distortions and alteration introduced by the opticalconfiguration, and to collect, store and combine a plurality ofsub-images corresponding to a plurality of regions of spatial frequencyspace to create a composite image, wherein the plurality of sub-imagesare formed as a result of adjustments to the first, the second, and thethird optical systems. In various embodiments, the third optical systemof the apparatus 1800 can include one of the third optical systemconfigurations shown in FIGS. 5A-5E.

In various embodiments, the grating 1864 profile can have an impact onthe extraction efficiency. In some embodiments, the grating 1864 canhave a sinusoidal profile. A sinusoidal grating has components in itsFourier transform only at ±1/d. In other embodiments, the grating 1864can have a rectangular profile. A rectangular grating has many moreFourier components that can lead to coupling of additional scatteredimage plane waves across the interface. For equal line: space grating,the second order Fourier coefficient (@±2/d) vanishes, although forsufficiently deep gratings, comparable to the wavelength, additionalcoupling terms can arise. The third order terms (@±3/d) are alwayspresent for rectangular grating profiles. This can give rise to multiplecoupling orders which can lead to artifacts in the sub-images. In somearrangements, this is not an issue because of the spatial separation ofthe scattered spatial frequency information at the bottom of thesubstrate (as can be seen in FIG. 18). In this case, the bottomsubstrate plane is separated from the object plane and the differentspatial frequency components, propagating at different angles, haveseparated to some extent by the time they reach this plane. If thethickness of the substrate 1825 is significantly larger than the fieldof view (illuminated aperture at the image plane), this separation canbe large enough to avoid issues associated with higher order coupling atthe bottom surface extraction grating. Thus, there is engineeringtrade-off in choosing the thickness of the substrate 1825, theseparation is better if it is thicker, but the phase distortions areincreased.

Alternative collection schemes can include using one or more prisms1974, as shown in FIG. 19. In some embodiments, the prism 1974 can befabricated as part of the substrate 1925. In other embodiments, indexmatching fluid 1972 can be used.

In certain embodiments apparatus 1800 for microscopy can also include atleast one known reference object to cover a small part of the imagefield.

According to various embodiments, there is a method for microscopy byillumination through a substrate. The method can include providing anobject 1820 disposed over a first side of a planar substrate 1825, thesubstrate characterized by a homogeneous refractive index (n_(sub)) anda surface normal 1826 such that the object 1820 is separated from thesubstrate 1825 by a distance of no more than about ≦λ. The method canalso include providing at least one grating 1864 on the side of thesubstrate 1825 opposite the object plane 1822, each grating 1864characterized by a position, a period, a depth, and a grating profile,wherein each of the gratings 1864 can further scatter reflected wavesresulting from the coherent illumination of the object into propagatingwaves in the medium below the substrate. The method can further includeproviding a first optical system disposed to provide a substantiallycoherent illumination of the object plane, the illuminationcharacterized by a wavelength λ and a radius of curvature and disposedat one of a plurality of incident wave vectors from about 0 to about2πn_(sub)/λ with respect to a surface normal of the substrate and at aplurality of azimuth angles spanning from about 0 to about 2π. Themethod can also include providing a second optical system 1830 having anoptical axis 1836 disposed at one of a plurality of center wave vectorsfrom about 0 to about 2π/λ with respect to the surface normal 1826,wherein the second optical system 1830 includes at least one grating1864 on the second side of the substrate 1825 and is characterized by anumerical aperture (NA). The method can further include providing athird optical system disposed in an optical path of the first opticalsystem to provide interferometric reintroduction of a portion of thecoherent illumination (reference beam) into the second optical system1830, wherein each of an amplitude, a phase, a radius of curvature andan angle of incidence of the reference is adjusted as required such thata corrected reference wave is present at the image plane of the secondoptical system 1830. The method can also include providing an electronicimage device disposed at an image plane of the second optical system1830 that responds linearly to the local optical intensity and transfersthe local optical intensity map across the image plane (a sub-image) toa signal processor device in electronic form, providing a signalprocessor that receives the electronic form of the sub-image,manipulating each of the plurality of sub-images using the signalprocessor to correct for distortions and alterations introduced by theoptical configuration, and combining the plurality of sub-images into acomposite image to provide a substantially faithful image of the object.In various embodiments, the method can further include one or moreprocesses of subtraction of dark field images, subtraction of backgroundimages, shifting of spatial frequencies in accordance with the opticalconfiguration, and elimination of one or more overlapping coverages ofthe frequency space wherein the elimination operations can be performedeither in the optical systems or in the signal processing. In someembodiments, the method can also include selecting regions of spatialfrequency space to provide a more or less faithful image of the objectin the object plane.

For various IIM configurations shown in FIGS. 3, 12, and 18, thecoherence length >>sample (object) dimensions. The He—Ne laser has along coherence length of many cm, and this makes the experimentalarrangement simpler, as it is not necessary to critically match theinterferometer lengths between the objective arm and the zero-orderreinjection arm. However, it does increase spurious speckle effectsassociated with stray light and multiple reflections from variousoptical surfaces in the set-up. These effects can be mitigated bychoosing a source with sufficient coherence for the IIM measurements,but insufficient coherence for Fabry-Perot effects, e.g. between thefront and back sides of the substrate or between the substrate and theobjective entrance surface. Since, these dimensions are very different,μm scale for the sample to several mm for the substrate thickness andsubstrate to objective distance, it is possible to minimize unrelatedFabry-Perot effects while retaining all of the resolution of IIM.

Tiling of Frequency Space

In general, the spatial frequency location of the informationcorresponding to a specific angle of illumination (includingillumination through the substrate) and angle of collection (θ)corresponds to

${\overset{\rightarrow}{k}}_{scatter} = {{\frac{2{\pi (n)}}{\lambda}\sin \; \theta_{illumination}{\hat{e}}_{illumination}} - {\frac{2\pi \; n_{pp}}{\lambda}\sin \; \theta_{scattered}{\hat{e}}_{scattered}}}$

In the above equation, (n) in the first term is adjusted as appropriate,for example, for illumination in air, n_(pp)=1 while for illumination(evanescent) through a substrate, n=n_(sub)=1.5 for glass. In keepingwith the notation established above, θ_(scattered) is the angle in thesubstrate and so the factor n_(sub) is appropriate; a grating can beused to shift the spatial frequencies into the air propagation bandpassas necessary.

Both angles as well as the pitch of any gratings can provide someflexibility in the tiling of frequency space, i.e. in choosing theregions of frequency images into a complete image. The maximum spatialfrequency, k_(max)=2πf_(max)=2π(2n_(sub)/λ) is obtained when both anglesare close to 90°. Since a half pitch can be resolved, this leads to anAbbe resolution limit of λ/4n_(sub). The optimum strategy is patterndependent, for example, for Manhattan geometry structures with edgesconfined to a rectangular grid, often found in integrated circuits, itis important to capture the frequency information along the axes and ofless consequence to capture the information away from the axes where theFourier transform of the pattern has lower spectral intensity. In theexamples shown in FIGS. 19A and 19B, only one principal axis isconsidered, but the generalization to complete coverage isstraightforward.

FIGS. 20A and 20B show two alternate embodiments for providing coveragefrom f_(x)=0 to f_(x)=2n_(sub)/λ with a fixed objective NA. Thefrequency space coverages shown in FIGS. 20A and 20B are designed for amaximum spatial frequency of 3/λ (2n_(sub)/λ), and an objective of NA of0.65. In FIG. 20A, a minimum number of sub-images are used. The centralsmall circle corresponds to conventional, normal incidence coherentillumination with the radius of the circle being about 0.65/λ. The nextsub-image is taken with off-axis illumination through the substrate atan angle of about 53°; this corresponds to effective NA_(ill) of about1.2. The scattered light can be detected either from the top (throughair) or through the substrate, the collection geometry can be similar tothat shown in FIG. 18, except for the illumination direction. Azero-order (interferometric reference) beam can be used in IIM toprovide access to the essential phase information as well as to allowunambiguous assignment of the directly measured spatial frequencies. Thesquare law, intensity detection process restores both the complexconjugate frequencies within the symmetrically located dotted circle inthe figure. A third sub-image can be taken with grazing incidenceillumination through the substrate, and with higher scattered spatialfrequencies with the use of a grating of period λ/0.8 for extraction asin FIG. 18. Again, the complex conjugate spatial frequencies arerestored by the square-law detection process. For a Manhattan geometryobject, a similar set of sub-images in the orthogonal direction can beused, for a total of five sub-images; arbitrary structures may requireadditional sub-images to fill all of frequency space.

FIG. 20B shows a second tiling embodiment, using four sub-images, butprovides more robust coverage of frequency space (fewer missed spatialfrequencies in the regions where the circles abut). The central circleis the same as in the previous example, illumination at normal incidenceand collection with a conventional 0.65 NA optical system. The secondinnermost set of circles corresponds to illumination at grazingincidence in air (NA_(ill)˜1). The next innermost set corresponds to thesame illumination condition, but to collection through a glasssubstrate. The final outermost set of circles corresponds toillumination at grazing incidence through the substrate and collectionwith the same grating to allow high spatial frequencies (collection oflight scatted at angles beyond the critical angle in the glass). Thedisclosed exemplary embodiments provide an example of the flexibilityoffered by the IIM process. The choice of tiling strategy will depend onthe object to be imaged. In general, it is best not to put a collectionboundary in a region of high spectral intensity to minimize Gibbs effectoscillations of the observed sub-Image structure. In addition, thestrength of scattered spatial frequency components in the regionsbetween the circles will be a factor in selecting an IIM tilingstrategy.

It should be noted that the tiling with circular regions is not arequirement, but is convenient as a result of the symmetry of opticallenses. In some cases, a square aperture, which can be provided eitheroptically or electronically during the sub-image manipulations, canprove advantageous. In particular, a square aperture can be configuredto provide more robust coverage at the boundaries between sub-images(e.g. two squares can match along the line, while two circles can onlytouch at a point). The tilings in FIG. 19B show some overlap regions.Several strategies are available for dealing with the multiple countingin frequency space that these overlaps imply. The simplest is just toremove the double counting in the computation of the sub-imagecombination. Alternatively, a graded transfer function can be applied inthe region of the overlap to minimize artifacts from imperfectcancellation of Gibbs effect oscillations in the two sub-images. Thesimplest approach is to calculate the Fourier transform of thesub-image, apply appropriate filters and inverse transform back to realspace. The apparatus of Image signal processing is very rich, and manyof its techniques can be applied to this image reconstruction problem.

For arbitrary images, where a-priori information on likely orientationsand spatial frequency content is not available, for example biologicalspecimens, additional sub-images can be used in order to get a morecomplete coverage of spatial frequency space. An example of coveringmost of spatial frequency space is given in FIG. 21. This consists of 13sub-images: the two off-axis sub-images shown in the top of FIG. 20A arerepeated with rotation angles of 45°, 90 and 135° (there is no need torepeat the low-frequency sub-image) for a total of 9 sub-images;additional high frequency sub-images at rotation angles of 22.5°, 67.5°,112.5°, and 157.5°, for a total of 13 sub-images, complete the coverageexcept for small regions near the outer periphery of frequency space. Itshould be noted that there are only three optical configurations;on-axis illumination (low frequency), middle frequency, and highfrequency, the remaining sub-images are obtained by a simple samplerotation. Furthermore, provision can be made for illumination throughthe substrate for the middle and high frequency coverage as the sampleis rotated.

The number of sub-images can be reduced by increasing the objective NA.As can be seen in FIG. 22, the number of sub-images for nearly fullcoverage is reduced to 5 for a NA=0.95, corresponding to a very high NAair-based objective. The specifics of the arrangement is that the lowfrequency sub-image is taken for normal incidence (NA_(ill)=0); each ofthe offset sub-images is at NA_(ill)+G/λ=2 which can be achieved withgrazing incidence illumination through the substrate along with agrating with a period of λ/0.5.

FIG. 23 provides two similar 1D tiling strategies for a siliconsubstrate (n_(sub)=3.6 at 1064 nm), one (vertical) for a 0.65 NA andanother (horizontal) for a 1.4 NA conventional immersion objective. Asmany as seven sub-images may be used to provide a complete coveragealong just one axis for the 0.65 NA, whereas only three are sufficientfor the large NA. The area of frequency space, and the required numberof sub-images for nearly complete coverage, increases as n_(sub) ² orn_(pp) ². Scaling from FIG. 20A suggests that as many as(3.6/1.5)²×13˜75 sub-images would be required for full coverage with the0.64 NA objective. This suggests that there will be great advantage inknowing something about the image and its spectral content. Onesituation where this is dearly possible is in the inspection of siliconintegrated circuits. The demands of manufacturable lithography at thenanoscale are forcing lithographers to restrict the range of patternsallowed in modern integrated circuits. This is often referred to aslithography “friendly” design, which in general is forcing the patternscloser to periodic grating patterns. In turn, a lithography “friendly”circuit is a microscopy “friendly” circuit with a limited range ofspatial frequencies, hence complete coverage of spatial frequency spaceis not required to reconstruct an image. Immersion lenses are notavailable at an NA corresponding to the refractive index of silicon(3.6). An available immersion lens designed for more modest NAs of ˜1.4can be used with the addition of gratings to couple the higher spatialfrequency light out of the substrate or plane parallel optical element.An issue with the very high NA immersion lens is that these lensestypically have a very short working distance, which in turn will requirea very thin substrate or plane parallel optical element, or a speciallydesigned objective.

In implementations, immersion advantages of IIM can be realized if theobject is in close proximity to a solid-immersion medium withillumination and collection through a plane-parallel optical element andcoupling this radiation to air by one or more gratings on the mediumsurface opposite the object. The plane-parallel optical element differsfrom the substrate, as discussed above, in at least one manner, which isthat the substrate functions, at least in part, to support the object.The plane-parallel optical element, as discussed further below, is ofsuch a size and thickness that it cannot alone function to support theobject. In implementations, a mechanical coupler can be used to supportthe object and optically couple the object with the plane-parallelplate. The plane-parallel optical element can be characterized by ahomogeneous refractive index (n_(pp)) and a thickness (t_(pp)), suchthat a distance of separation between the plane-parallel optical elementand the object can be within a wavelength of the light used toilluminate the object.

In implementations, the plane-parallel optical element can include highindex of refraction materials to function as a solid-immersion medium.When used in IIMM, the plane-parallel optical element can have athickness (t_(pp)) of about a fraction of the illumination wavelength,which allows use of strongly absorbing materials. By way of anon-limiting example, the wavelength can be chosen to be shorter thanthe band gap of the immersion material to take advantage of theincreasing refractive index within an absorption band. In such anexample, both the shorter wavelength and the larger index can extend theresolution will beyond those conventionally available with other knownlinear system microscopy implementations, even within the same λ/4n_(pp)linear-systems frequency-space limit, where n_(pp) is the refractiveindex of the plane parallel optical element.

IIM can be adapted for solid-immersion in a variety of ways. Forexample, a high-index plane-parallel optical element can be placed veryclose to the object (within range of the evanescent fields). Theplane-parallel optical element can be a multilayer structure or aconfined liquid and can be at least partially transparent at the IIMwavelength. Experiments conducted by the inventors have shown aresolution <λ/5 for IIM of high contrast for chrome-on-glass objects onplane-parallel optical element composed of glass by illumination throughthe evanescent fields of the plane-parallel optical element andconventional collection in air (including off-axis contributions). Forcompactness, this configuration is referred to below as half-immersion(e.g. the Illumination takes advantage of the higher wave vectors in theplane-parallel optical element, but the collection is limited toscattered wave vectors that propagate in air). For this configuration,the limiting optical resolution, in the Abbe sense of the smallestavailable half-pitch (HP) in the optical response, is λ/[2(n_(pp)+1)].

The resolution can be further extended by collection of theback-scattered light that propagates within the plane-parallel opticalelement beyond the angle for total internal reflection. This scatteringcorresponds to larger wave numbers and therefore to information onsmaller details of the sample. A grating can be used on the back side ofthe plane-parallel optical element, opposite the side or surface facingthe object or sample being imaged, to extract this scatteredinformation, making it accessible to the optical collection system.There are spatial-frequency-dependent distortions associated with thespreading of the information as it propagates across the plane-paralleloptical element and extends spatially beyond the field of view of thecollection system, and phase aberrations associated this extendedpropagation, which require a mechanism or protocol for transforming theimage from the laboratory frame to the image frame for combination withother sub-images. This mechanism or protocol can be used to compensatefor frequency and/or phase discrepancies that can result from theoptical configuration. The linear systems limit is a resolution ofλ/4n_(pp); resolution very close to this limit can be achieved in manycases, however with interrelated requirements on the field-of-view, thenumerical aperture, and the thickness and refractive index of theplane-parallel optical element.

As discussed herein, the relationship between these parameters and thenumber of required sub-images are quantified and the transformationprocedure for sub-images for deep sub-wavelength cases with fullimmersion, including single and multiple backside sub-images isdiscussed. The techniques described herein can use one or more gratingsdisplaced from the object by the thickness of the plane-parallel opticalelement to extract waves propagating in the plane-parallel opticalelement beyond the angle for total internal reflection and make themavailable for collection in free space.

In a non-immersion IIM optical arrangement, the result is the capture ofa sub-image consisting of a separate region of frequency space in eachsub-image. In air, the maximum angle of illumination can approach 90°,but is smaller in the plane-parallel optical element as a result ofrefraction at the air-plane-parallel optical element interface. Bycoupling into internal modes of the plane-parallel optical element,grazing incidence in the plane-parallel optical element can be achieved,increasing the resolution. IIM relies on recording and combiningsub-images to produce a final image that covers all of the essential,object dependent, parts of frequency space.

Using only a modest NA=0.4 lens at λ=633 nm and an object coupled to aplane-parallel optical element with refractive index n_(pp), ahalf-immersion imaging resolution technique with evanescent waveillumination is disclosed to a maximum spatial frequency of(n_(pp)+NA)/λ with the objective normal to the plane-parallel opticalelement (e.g. un-tilted objective; 170 nm features on a glassplane-parallel optical element with n_(pp)=1.5) and up to (n_(pp)+1)/λwith a tilted objective (150 nm features of arbitrary structures, whilethe theoretical limit of grating HP resolution is 126 nm). A gratingcoupler can be added to the side of the plane-parallel optical elementopposite the object to collect the spatial frequency information between(n_(pp)+NA)/λ and (n_(pp)+1)/λ as well as extending the spatialfrequency coverage towards 2n_(pp)/λ. Phase and amplitude matching ofthe sub-images can be achieved electronically using a reference objectcontaining spatial frequencies within each recorded sub-image.

The illumination and collection configurations for half-immersion andfull-immersion are shown in FIG. 24A. The illumination laser beam, fromthe first optical system as shown and described above in relation toFIGS. 3 and 4 can be coupled into the plane-parallel optical element2405 using one or more optical components including, but not limited to,a prism, a grating or end fire coupling, as shown in FIGS. 12 and 13 andthe object 2410 can be illuminated by an evanescent wave. Imagefrequencies up to (n^(pp)+NA)/λ can be captured with an second opticalsystem having an objective normal to the plane-parallel optical elementsurface (FIG. 24A, objective A 2420A), and frequencies up to(n_(pp)+1)/λ with tilt of the objective off of the optic axis (FIG. 24A,objective B 2420B). The evanescent waves from higher frequency contentof the object 2410 are coupled back into the plane-parallel opticalelement by the boundary conditions at the plane-parallel opticalelement-object interface, and for spatial frequencies between(n_(pp)+1)/λ and 2n_(pp)/λ propagate in the plane-parallel opticalelement 2405 at angles beyond the angle for total internal reflection.For a flat interface, the information at these spatial frequencies isnot accessible, but the scattered light can be decoupled by one or moregrating structures on the side of the plane-parallel optical element2405. The plane-parallel optical element 2405 can include a surface thatis opposite the object 2410 and can include the one or more gratingstructures 2415, each with grating structure having its owncharacteristic position, period, depth, and/or grating profile.Radiation can be redirected by the one or more grating structures to anobjective positionable on the opposite side of the plane-paralleloptical element then the sample (FIG. 24A, objective C 2420C). Thisoptical system (the required coherent reference beam is not shown) leadsto frequency aliasing as a result of the grating diffraction. While thiscan be corrected with the reference beam, it is usually preferable tooffset the sub-image spatial frequencies to lower intermediatefrequencies to reduce the pixel size and density requirement on thecollection system focal plane and restore the actual frequenciescomputationally before combining sub-images. In addition, there arephase errors (aberrations) associated with the collection system whichincludes partial propagation both in the high-index plane-paralleloptical element and in air. The treatment of these spatial frequency andphase corrections is discussed below.

The corresponding frequency space coverages achievable using theapparatus configurations of FIG. 24A are shown in FIG. 24B. Normalincidence illumination, and collection from the sample side is thetraditional coherent illumination configuration represented by thecircle 2405 with frequency space coverage to NA/λ. Illumination at anangle of 2NA/λ provides the offset circles 2410 with frequency spacecoverage to 3NA/λ. For a Manhattan geometry object, two sub-imagesproviding coverage in the x,y directions is typically used, additionalsub-images, indicated by the circles 2415 (at 45° to the principal x,yaxes) can be added for additional off-grid frequency space coverage. Theillumination and sample-side collection scheme of objective position Aof FIG. 24B allows increasing the spatial frequency coverage to(n_(pp)+NA)/λ (circles 2420). Collection with a tilted objective, asshown with objective position B of FIG. 24A, allows frequency spacecoverage to (n_(pp)+1)/λ. Finally, the plane-parallel optical elementside collection discussed in this contribution, objective position C ofFIG. 24A, extends the frequency space coverage to the linear systemslimit of 2n_(pp)/λ with a corresponding Abbe half-pitch of λ/4n_(pp).

FIGS. 25A-D show four example arrangements for the plane-paralleloptical element 2405 with respect to the object 2410 that can be usedwith optical arrangement shown in FIG. 24 according to implementationsof the present disclosure.

FIG. 25A shows the plane-parallel optical element 2405 as a membrane2505 having support members 2510 on at least a portion of the membrane2505, wherein the membrane 2505 is optically coupled with the object2515. In this arrangement, thickness of the membrane 2505 is typicallynot sufficient to fully physically support the object 2515. Therefore,the support members 2510 can function, at least in part, to add strengthor resiliency to the membrane 2505. The membrane 2505 can include afirst surface 2520, which is arranged on the side of objectives A and 8,and a second surface 2525, which can include one or more gratings 2530and is arranged on the side of objective C.

FIG. 25B shows the plane-parallel optical element 2405 as a superstrate2535. In this arrangement, the object 2515 can be positioned between asupport 2540, for example, a substrate, on one side and a superstrate2535 on the other side. The superstrate 2535 can be arranged on the sideof objective C and can include one or more gratings 2545. The lowerfrequency information, collected by objectives A and B can be extractedeither from the top surface 2515 of the superstrate 2535 or from thebottom surface of the substrate 2540. Substrate 2540 can be locallythinned (not shown) to reduce optical aberrations in the collectionsystem.

FIG. 25C shows the plane-parallel optical element 2405 as a multilayersuperstrate 2550. In this arrangement, which is similar to thearrangement shown in FIG. 25B, one or more materials can be used in themultilayer superstrate 2550, including superstrate layers 2550 a and2550 b, wherein each material can have a specific thickness and opticalcharacteristic, including different indexes of refraction. Although onlytwo layers are shown in the figure, more than two layers for thesuperstrate can be used. In this arrangement, the object 2515 can bepositioned between a support 2540, for example, a substrate, on one sideand a multilayer superstrate 2550 on the other side. The superstrate2550 can be arranged on the side of objective C and can include one ormore gratings 2545. The lower frequency information, collected byobjectives A and B can be extracted either from the top surface of thesuperstrate 2550 or from the bottom surface of the substrate 2540.Substrate 2540 can be locally thinned (not shown) to reduce opticalaberrations in the collection system.

FIG. 25D shows the plane-parallel optical element 2405 as a suspendedsuperstrate 2555. In this arrangement, which is similar to thearrangement shown in FIG. 25B, suspended superstrate 2555 can besupported above the object 2515 by one or more intermediate materials.For example, the one or more intermediate materials can be in a fluidform and each material can have a specific volume and opticalcharacteristic, including different indexes of refraction. In thisarrangement, the object 2515 can be positioned between a support 2540,for example, a substrate, on one side and a suspended superstrate 2555on the other side. The superstrate 2555 can be arranged on the side ofobjective C and can include one or more gratings 2545. The lowerfrequency information, collected by objectives A and B can be extractedeither from the top surface of the superstrate 2545 or from the bottomsurface of the substrate 2540. Substrate 2540 can be locally thinned(not shown) to reduce optical aberrations in the collection system.

In each of the arrangements shown in FIGS. 25A-D, the separationdistance between the plane-parallel optical element and the object 2510is within the wavelength of the light used to illuminate the object2515.

IIM requires reintroducing a coherent zero-order reference at the imageplane (e.g. constructing an interferometer around the objective lens) torecord the amplitude and phase of the spectral frequencies in thesub-images. The intensity, angle and phase of the reference beam have tobe chosen to match all sub-images to the on-axis image. A referenceobject is used to cover a small part of the FOV in order to determinethe correct amplitude ratio, frequency shift and phase. These offsetfrequencies can then corrected in the image processing before thesub-images are combined.

In the description that follows, elements including ray tracing (lookingat the propagation of scattered rays corresponding to specific spatialfrequencies) and Fourier optics (based on “infinite” plane wavepropagation) are both presented. To bring these two concepts together,“wave packets” will be considered with center spatial frequencies thatcorrespond to the direction of propagation and with a spatial extentthat corresponds to the field-of-view, which is assumed to be muchlarger than the individual scattering objects within the field, but muchsmaller than the diameter of the lens. This corresponds to a broadeningin the pupil plane and Fourier planes from the delta functionsassociated with plane waves to diffraction patterns corresponding to thefinite field of view.

In embodiments, additional scattered information can be collected atspatial frequencies beyond (n_(pp)+NA)/λ by collection from the backside of the plane-parallel optical element using one or more gratings toredirect this information into an objective lens. FIG. 26 shows anexample geometry that shows access to collection high frequenciespropagating in the plane-parallel optical element that correspond tosmall features. As can be seen from the geometry of FIG. 26, the spatialfrequency coverage of each sub-image depends on the thickness andrefractive index of the plane-parallel optical element as well as on thefield-of-view (FOV) and NA of the objective lens. For thickerplane-parallel optical elements, the relevant information is spreadacross a wider area requiring a larger FOV. This may require multiplespatially displaced sub-images to extract all of the information (asynthetic FOV). If the available information extends beyond the 2NA/λbandwidth of the collection optics, multiple gratings are required (asynthetic aperture). The minimum collected spatial frequency (angle atin FIG. 26) sets the period d of the extraction grating:

$\begin{matrix}{d = \frac{\lambda}{{n_{pp}\sin \; \alpha_{1}} + {NA}}} & (9)\end{matrix}$

If this frequency equals the maximum available from half immersionwithout a tilted objective, (n_(pp)+NA)/λ, then:

$\begin{matrix}{d = {\frac{\lambda}{2\; {NA}}.}} & (10)\end{matrix}$

This takes a scattered wave in the plane-parallel optical elementcorresponding to

k _(α) ₁ =k ₀ NA=n _(pp) k ₀ sin α₁ =n _(pp) k ₀ sin └ sin⁻¹(NA/n_(pp))┘   (11)

into a wave propagating in air at an angle −sin⁻¹(NA). Here, k₀=2π/λ.Note that provided NA >0.33, higher diffraction orders from the gratingare outside the NA of the collection optics and do not interfere withthe image; an NA=0.4 is considered in the modeling. Over the range ofspatial frequencies collected in each sub-image the diffractionefficiencies are roughly constant, thus allowing intensity compensationby sub-image matching procedures. This technique tends to be free of thecomplications connected with multiple diffraction orders from gratingsencountered by other approaches. In embodiments, the gratings canprovide extraction of information out of the immersion media but notdiffraction of near-field high-spatial frequency components directlyfrom the object. There can be variations in diffraction efficiency asthe various higher order beams, in both the plane-parallel opticalelement and in air, switch from evanescent to propagating waves. Thesecan be dealt with empirically by adjusting the amplitudes of therelevant portions of each sub-image independently, either by physicallyrestricting the collection NA appropriately, or by separately addressingthe regions of the sub-image electronically.

Progressively higher spatial frequency components impinge on the gratingat larger horizontal displacements from the object and are diffractedinto increasing angles, until the scattered beam at a displacement ofb+F from the object centerline is diffracted at to an angle of +θ inair. The distance F corresponds to the FOV of the objective lens, whichcan be taken as focused on the grating surface, or to the width of thegrating if it is smaller than the FOV.

Provided θ≦sin⁻¹(NA), the entire spread of scattered light incident onthe grating is collected by the objective lens. From the geometry ofFIG. 25, several important relationships are readily derived:

$\begin{matrix}{{{{\sin \left( \alpha_{1} \right)} = {\frac{NA}{n_{pp}} = \frac{b}{\sqrt{b^{2} + t^{2}}}}};{b = {t\left\lbrack {\left( \frac{n_{pp}}{NA} \right)^{2} - 1} \right\rbrack}^{{- 1}/2}}}{and}} & (12) \\\begin{matrix}{{\sin \left( \alpha_{2} \right)} = \frac{b + F}{\sqrt{\left( {b + F} \right)^{2} + t^{2}}}} \\{= \left\lbrack {1 + \left( \frac{b + F}{t} \right)^{- 2}} \right\rbrack^{{- 1}/2}} \\{= \left\{ {1 + \left\lbrack {\frac{F}{t} + \left( {\left( \frac{n_{pp}}{NA} \right)^{2} - 1} \right)^{{- 1}/2}} \right\rbrack^{- 2}} \right\}^{{- 1}/2}}\end{matrix} & (13)\end{matrix}$

and the corresponding minimum half pitch is:

$\begin{matrix}{{HP}_{\min} = {{MAX}\begin{Bmatrix}{\frac{\lambda}{2\left( {n_{pp} + {3\; {NA}}} \right)};} \\\frac{\lambda}{2\; n_{pp}\left\{ {1 + \left\{ {1 + \left\lbrack {\frac{F}{t} + \left( {\left( \frac{n}{NA} \right)^{2} - 1} \right)^{{- 1}/2}} \right\rbrack^{- 2}} \right\}^{{- 1}/2}} \right\}}\end{Bmatrix}}} & (14)\end{matrix}$

The upper expression (Eq. 12) is valid when the full NA of the objectivelens is filled by the diffracted beams from the grating, e.g. thegrating width F, and the optical FOV and NA are such that θ≧sin⁻¹(NA).If the angular spread is restricted by the field of view, orequivalently by the width of the grating, the lower expression (Eq. 13)pertains. An additional constraint is that 3NA<n_(pp), since onlyspatial frequencies that can propagate in the plane-parallel opticalelement can be collected. The limiting behavior of HP_(min) is readilyevaluated from this expression. For small NA where the full angularwidth of the lens is filled, the upper expression (Eq. 12) applies. Forall interesting cases, NA/n_(pp)<<1; that is the lens NA is much lessthan the refractive index of the immersion medium. For large fields ofview or thin plane-parallel optical elements, F/t>>NA/n_(pp),

$\left. {HP}_{\min}\rightarrow{\frac{\lambda}{n_{pp}\left\lbrack {4 - \left( {t/F} \right)^{2}} \right\rbrack}.} \right.$

Thus, HP_(min) is always larger than the optics linear systems limit.The upper limit in Eq. 13 takes over before this result; thus the NA ofthe lens is filled in just a single sub-image. Additional gratings atsmaller pitches of λ/2(i+1)NA [i=1, 2, 3, . . . ] allow access to higherspatial frequency components of the image up to the linear systems limitof λ/4n. In the opposite limit, NA/n_(pp)<<1 and F/t<<NA/n_(pp),

$\left. {HP}_{\min}\rightarrow{\frac{\lambda}{2\left( {n_{pp} + {NA} + \frac{n_{pp}F}{t}} \right)}.} \right.$

The resolution is always somewhat improved over the starting point ofhalf-immersion with the collection system optical axis normal to theobject plane. In this case the linear systems limit of λ/4n_(pp) can beapproached with a synthetic FOV, e.g. multiple sub-images with thecollection optical system displaced to collect the higher spatialfrequencies that are lost by the limited FOV with the same grating, andagain, with multiple gratings (synthetic aperture), it is possible toextend the resolution close to the λ/4n_(pp) limit, as long assignal/noise ratio is sufficient to enable sub-image reconstruction intoa full image.

Resolution (HP) restrictions as a function of plane-parallel opticalelement refractive index for NA=0.4, 0.8, 1.2, fixed field of view (F=32μm) and plane-parallel optical element thickness (t=50 μm) obtained fromEq. 14 are shown in FIG. 27. There is a point of transition on eachcurve (solid to dotted). The solid lines correspond to the upperexpression of Eq. 14; the dotted lines to the lower form. In the dottedregion additional sub-images are required to synthesize a larger FOV.Once the lens NA is filled, an additional grating is required to extracthigher spatial frequency information and alias it into the lens NA, e.g.to synthesize a larger NA.

Exemplary combination of restrictions induced by plane-parallel opticalelement properties and synthetic aperture (multiple of NA=0.4) for afixed field of view (F=32 μm) with varying plane-parallel opticalelement thickness are shown in FIG. 28. One of ordinary skill in the artwould appreciate that the combination of restrictions tends to change asthe numerical aperture (NA) changes. The curves correspond toplane-parallel optical element thicknesses of 10, 30, 100 and 300 μmwith break points denoted by the transitions from dashed to dotted linesby curves of synthetic NA restrictions. Here, λ/[2(n_(pp)+3NA)]corresponds to upper part of Eq. 14. The restrictions λ/[2(n_(pp)+5NA)]and λ/[2(n_(pp)+7NA)] appear by synthetic aperture extension with 1 and2 additional aperture intervals along each spatial direction usingadapted gratings for each interval as described above.

It can be inferred from FIGS. 27 and 28 that, for a single sub-image, asmall NA optical system can give useful resolution extensions only formaterials with low index of refraction. In order to reach highresolution using materials with high n_(pp), either additionalsub-images using multiple gratings or an objective with higher NA isneeded. A larger FOV objective enhances the resolution but typically isassociated with lower NA, which again requires additional sub-images. Acompromise between FOV and NA has to be found for the chosenplane-parallel optical element thickness and index of refraction tominimize the total number of sub-images. These models do not include theimpact of a finite signal-to-noise ratio (S/N). As the signal becomesmore dispersed with thicker plane-parallel optical elements, the S/Ndecreases and stochastic contributions to the image become moresignificant limiting the ability to accurately combine the sub-imagesand construct a composite image.

Initial experiment were conducted using a 1-mm thick glassplane-parallel optical element optically coupled to a second 1-mm thickmicroscope slide with a metal decoupling grating of period 560 nm. Thusthe total thickness (object to grating) is 2 mm. The results showed thepossibility of resolution of a periodic structure. The image consists ofa repeated pattern of several parallel lines with a spacing of 240 nmwithin a trapezoidal envelope. The pattern is repeated at a spacing of3.6 μm in both directions. A SEM image is shown in FIG. 29A. Thex-direction high frequency image was recorded and is shown in FIG. 29B.The high frequency image contains much of the Information about theoriginal pattern: the repeated pattern is evident as is the clusteringof lines in each repeat unit. However, the image is distorted due to thegeometry of propagation in the plane-parallel optical element [FIG.24(b)] and requires a restoration procedure before the proper image canbe recovered. Clearly there are fewer dusters at the same transversescale (3 vs. 4) in the distorted image, the relative spacing between theline clusters is changed and there are additional lines in the dusters,though the line pitch remains the same

The distortion of the image as a result of the propagation in the anddepends on the optical path in the plane-parallel optical element, e.g.on the plane-parallel optical element refractive index and thickness.The optical configuration was shown in FIG. 24, with the collection lensfocused onto the grating surface. Since an aberration-free opticalsystem has no phase error between conjugate planes, e.g. the gratingsurface and the camera focal plane, the only phase variations needed tobe considered are for propagation in the plane-parallel optical element(FIG. 26). For analytical simplicity, a one dimensional case isconsidered; the calculations are readily extended to two dimensionalobjects. Let L and L₀ be optical paths of an arbitrary and of thecentral ray in the plane-parallel optical element, α and α_(c) are theangles of the corresponding rays to the plane-parallel optical elementnormal. θ is the angle of the arbitrary ray to the optical axis afterdiffraction from the grating and exiting the plane-parallel opticalelement (the ray must be captured by the objective in air and forconvenience is shown as a marginal ray). The angle α₀ of the ray in theplane-parallel optical element which is redirected along the opticalaxis in air is:

$\begin{matrix}{{\sin \; \alpha_{0}} = {\frac{\lambda}{n_{pp}d} = \frac{2\; {NA}}{n_{pp}}}} & (15)\end{matrix}$

The marginal ray inclined at the angle α₂ to the normal in theplane-parallel optical element and an angle θ in air after scattering bythe grating is described by:

$\begin{matrix}{{\sin \; \alpha_{2}} = {\frac{1}{n_{pp}}\left( {\frac{\lambda}{d} + {\sin \; \theta}} \right)}} & (16)\end{matrix}$

Then the path lengths in the plane-parallel optical element are:

$\begin{matrix}{L_{0} = {\frac{t}{\cos \; \alpha_{0}} = {\frac{t}{\sqrt{1 - {\sin^{2}\alpha_{0}}}} = \frac{t}{\sqrt{1 - \left( \frac{\lambda}{n_{pp}d} \right)^{2}}}}}} & (17) \\{L = {\frac{t}{\cos \; \alpha_{2}} = {\frac{t}{\sqrt{1 - {\sin^{2}\alpha_{2}}}} = \frac{t}{\sqrt{1 - \left\lbrack {\frac{1}{n_{pp}}\left( {\frac{\lambda}{d} + {\sin \; \theta}} \right)} \right\rbrack}}}}} & (18)\end{matrix}$

and the phase difference between the arbitrary ray and the central rayis

$\begin{matrix}{{\Delta\phi} = {{\phi - \phi_{0}} = {{\frac{2\pi \; n_{pp}t}{\lambda}\left\lbrack {\frac{1}{\sqrt{1 - \left\lbrack {\frac{1}{n_{pp}}\left( {\frac{\lambda}{d} + {\sin \; \theta}} \right)} \right\rbrack^{2}}} - \frac{1}{\sqrt{1 - \left( \frac{\lambda}{n_{pp}d} \right)^{2}}}} \right\rbrack}.}}} & (19)\end{matrix}$

The rays in FIG. 26 are k-vectors of the plane waves propagating atangles θ corresponding to the image spatial frequencies f_(x). So, thephases at each spatial frequency can be corrected in Fourier space usingthe distortion phase function provided by the 2D generalization of Eq.19. Clearly that distortion phase function (Eq. 19) provides only arelative phase correction. The constant term (the phase shift introducedby the central ray optical path) will be automatically corrected laterby the sub-image phase-matching procedure required in IIM, since thisconstant term is indistinguishable from arbitrary constant termintroduced by the phase of the reference arm of the Mach-Zehnderinterferometer inherent in IIM.

Simulation of the impact of this phase distortion on the image withnested-L structure and a delimited grating with CD=120 nm (FIG. 30A) isshown in FIG. 31. The high spatial frequency (between NA/λ and 3NA/λ)filtered image of the model is shown in FIG. 308. The image is expanded,e.g. additional features appear on both sides of object due to lack ofcompensation in these regions as a result of the optical bandwidthlimit. This is just the familiar Gibbs effect associated with an abruptcut-off in frequency space. The high frequency image after theapplication of the phase aberrations for a plane-parallel opticalelement thickness of 1 μm [5 μm] is shown in FIG. 31A [FIG. 31B](crosscut FIG. 31C) [FIG. 31D]. There is additional walk-off of theintensity vs. position as a result of the spreading of the imageintensity. The reason for this spread is the progressive walk-off ofhigher spatial frequency components (phase distortions) as theypropagate across the plane-parallel optical element. Here, theadditional features appear non-symmetrically to the illumination side.Also, unlike the Gibbs effect, no information is lost in general. Thestep from FIG. 308 to FIGS. 31A and 31B are completely deterministic andis easily inverted by taking the Fourier transform of the laboratoryframe image, applying the inverse of Eq. 19 and retransforming back tothe image frame providing all information is captured and there are noS/N limitations. The spatial extent of the image spectrum expands withincreasing plane-parallel optical element thickness (compare FIGS. 31Aand 31B). The intensity spread extension beyond the objective field ofview leads to the loss of information which results in reduction of theimage quality after restoration. This information can be accessed with asynthetic FOV, e.g. shifting the objective lens to acquire additionalsub-images with an extended grating at the same pitch.

Without shifting the objective lens, the loss of information isequivalent to the reduction of captured range of frequencies(NA_(pp)<NA) for a single sub-image, which is a function of the FOV. Toevaluate this degradation of the image bandwidth in a single image,consider again FIG. 26, but now in a configuration where the grating ischosen so that a particular HP_(c) is along the optical axis, e.g. fixthe optical axis (center) frequency rather than the low-frequencymarginal ray. The dependence of the angular bandwidth, 2NA_(pp), versusthe FOV is easily to obtain from FIG. 26. The FOV (F) normalized to theslab thickness is:

$\begin{matrix}{\frac{F}{t} = {{\tan \; \alpha_{2}} - {\tan \; \alpha_{1}}}} & (20)\end{matrix}$

On the other hand the marginal angles for a particular NA_(sub) can bewritten as function of an angle sin α_(c) of the center frequency,corresponding to the chosen HP_(c).

sin α₂=sin α_(c) +NA _(pp)  (21)

and

sin α₁=sin α_(c) −NA,  (22)

where, for an illumination angle sin β:

$\begin{matrix}{{{\sin \mspace{11mu} \alpha_{c}} + {\sin \mspace{11mu} \beta}} = \frac{\lambda}{2n_{pp}{HP}_{c}}} & (23)\end{matrix}$

Combining Eqs. 19-21 gives an implicit relation for the optical systemparameters

$\begin{matrix}{\frac{F}{t} = {\frac{{\sin \mspace{11mu} \alpha_{c}} + \frac{{NA}_{pp}}{n_{pp}}}{\sqrt{1 - \left( {{\sin \mspace{11mu} \alpha_{c}} + \frac{{NA}_{pp}}{n_{pp}}} \right)^{2}}} - \frac{{\sin \mspace{11mu} \alpha_{c}} - \frac{{NA}_{pp}}{n_{pp}}}{\sqrt{1 - \left( {{\sin \mspace{11mu} \alpha_{c}} - \frac{{NA}_{pp}}{n_{pp}}} \right)^{2}}}}} & (24)\end{matrix}$

This dependence shown in FIG. 31 for several HP_(c) normalized by n andλ

$\left( {g = \frac{n_{pp}{HP}_{c}}{\lambda}} \right)$

allows us to define NA_(pp) of each sub-image and to estimate the numberof sub-images which are necessary to cover the of the available spatialfrequency space (along a specific direction).

It can be seen from FIG. 32 that, in order to prevent the loss ofinformation, an objective with a bigger FOV or additional spatiallyshifted sub-images to build a synthetic FOV is needed. These conclusionsare qualitatively the same as those drawn from FIGS. 27 and 28.

Examples of images shown in FIGS. 31A-31D restored using a FOV of 16 μmare shown with corresponding crosscuts 3005 in comparison with theundistorted image 3205 and differences between the restored and filteredcrosscuts in FIG. 33. FIG. 33A is obtained from FIG. 31A and FIG. 33Bfrom FIG. 31B. It is clear that the sub-image in FIG. 31A for a 1 μmthick plane-parallel optical element is extended less than the sub-imagein FIG. 31B for a 5 μm thick plane-parallel optical element and thequality of restored image in FIG. 33A is higher than in FIG. 33B.Extension of recorded field of view to 32 μm for the image in FIG. 31Bimproves the quality of restored image [FIG. 33C], showing the complexinterrelationships between the resolution, FOV, NA, plane-paralleloptical element thickness and the refractive index.

For an additional perspective on the ability to restore these images,the restored images with different HP were compared with the filteredhigh frequency images using a mean square error (MSE) metric. A simpleten-line grating pattern was chosen for MSE analyses (inside of thesquare 3210 of FIGS. 33A-C) and normalized to a gray field (FIG. 34).The curves of MSE versus HP for a λ=633 nm, n_(sub)=1.5 plane-paralleloptical element thicknesses of 0.5, 1, 3, 5, 10 μm and a restoration FOVof 32 μm are shown. For a comparable MSE procedure, it is important tohave the spectral content of the image filtered similarly. Thus, it isensured that the center frequency at the HP always passes through thecenter of the collection objective, as in the derivation of Eq. (24).

These calculations were carried out from the theoretical limitλ/4n_(pp)=0.106 μm to the half immersion limit λ/(n_(pp)+1)=0.126 μm(λ=633 nm, n_(sub)=1.5). The MSE drops as image becomes resolvable. Asexpected, the distortion (expansion of the frequency content across thedetection plane) of image features is lower in thinner films whichallows higher resolution with a smaller FOV.

The same models were used for plane-parallel optical elements withdifferent refractive indices in order to evaluate possible resolvable HPwith MSE=3% for plane-parallel optical element thicknesses of 1-, 5-,and 10-μm. The results are summarized in FIG. 34, where the resolvableHP versus refractive index are shown. The lower black dashed curve λ/4nis theoretical limit of full immersion resolution, the black upperdashed line λ/2(n_(pp)+NA) is the half-immersion limit with an un-tiltedobjective.

The modeling of image reconstruction represented in FIG. 35qualitatively confirms the results obtained by investigation oftheoretical resolution limit (FIG. 28). The image resolution depends onthe optical system and plane-parallel optical element properties (NA,FOV, t_(pp) and n_(pp)). The achievable resolution scales inversely withthe plane-parallel optical element index of refraction. Plane-paralleloptical element thicknesses greater than several times the FOV result inexperimental difficulties, both in registration and in lowered signalintensity leading to S/N issues.

The present techniques of the IIM configuration, as discussed above,with a slab of high refractive index material can be used as aneffective solid-immersion medium to enhance the resolution up to thelinear systems resolution limit of λ/4n_(pp). Phase distortions of highfrequency sub-images are inherent in the geometry of beam propagation inthe immersion slab, requiring a phase restoration procedure. Theresolution in this configuration depends not only on the objective NAand FOV, but also on the captured part of the spectral information whichis also a function of immersion slab refractive index and thickness. Thecriteria for evaluation of the ultimate HP limits for differentimmersion slab parameters and system field of view have been provided.The estimation shows that the minimum thickness of the immersion slaband the maximum field-of-view of the optical system should be chosen toachieve the highest resolution with the smallest number of sub-images.

Embodiments of the present disclosure allow a regime for IIM notachievable with conventional approaches. Using very thin plane-paralleloptical elements (or overlayers) and thereby restricting the propagationdistance, higher absorption can be tolerated, allowing the use ofshorter wavelengths. Then the resolution can be improved by two factors:the shorter wavelength; and the higher index of refraction within anabsorption band. The present approaches provide resolutions that are notavailable to solid immersion microscopy approaches as a result of theneed for a thick high-index solid immersion lens.

Table I provides calculated resolutions for several microscopytechniques and compares their practical resolution achievements fordifferent λ with a silicon immersion plane-parallel optical element.

TABLE I Wavelength dependent resolution on Si plane-parallel opticalelement for different techniques Wavelength (nm) 1064 704 633 488 430 Siproperties Si refractive index 3.56 3.77 3.9 4.37 4.91 Si 1/e length(μm) 1070 6.5 3.5 0.98 0.31 Double pass transmission 0.99 0.86 0.74 0.360.04 (0.5 μm) Alternative approaches Annular illumination 205 135 122 9483 (NA = 1.3) λ/4 NA SIL λ/4n (thick lens does 75 — — — — not allowmaterials with loss) IIM λ/4 (no immersion) 266 176 158 122 108λ/[2(n_(pp) + 1)] 115 74 65 45 36 (half immersion) λ/4n_(pp) (fullimmersion) 75 47 41 28 22Annular illumination using the ˜2× resolution advantage of off-axisillumination can be combined with immersion techniques (current resultsare with liquid immersion and an NA=1.3). However this requiresalignment between two specialized high NA, small FOV objectives which isa challenging task. Even ignoring the fact that usually there is atradeoff between the FOV and the NA, such objectives cannot usematerials with significant losses, as a result of the requiredmacroscopic optical thicknesses.

Solid immersion lenses (SIL) provide a relatively cost-effectivesolution for increasing NA by a combination of standard objective withsection of high index refraction sphere as solid immersion media. Thismethod has shown good resolution (to 145 nm using a Si SIL at 1.2 μm)but again can only be used with relatively long wavelengths since thesphere section (which in practice is close to a hemisphere) requiresessentially lossless materials. To the contrary, IIM can provide up tofew tens of nanometers resolution with immersion media such as siliconat visible (red to green) wavelengths while retaining the full field ofview, large working distance, depth of field, and low-cost of low NAobjectives.

Other materials coupled with wavelengths in proximity to a materialband-gap in combination with our method can also provide excellentresults. Some possible wavelength/material combinations to explore areshown in Table II.

TABLE II Examples of possible combinations of materials and wavelengthfor enhanced resolution λ (nm) λ/4 (nm) immersion λ/4n_(max) 633 158 48(n_(pp) = 3.3, GaP) 40 (n_(pp) = 4.0 Si) 488 122 50 (n_(pp) = 2.45, GaN)193 48 27 (n_(pp) = 1.8, Photoresist) 23 (n_(pp) = 2.1, Garnet) 19(n_(pp) = 2.6, Si₃N₄)

Thus IIM can be very useful for imaging small features using thinimmersion slab with high n_(sub) where resolution approaches that of aSEM with a simple inexpensive technique that is applicable in a range ofenvironments including air and water.

Imaging interferometric microscopy techniques as described above aresensitive to the optical refractive index variation of the objectmaterials and do not contain any material specific information. Imaginginterferometric microscopy can be applied to get material and chemicalinformation using coherent anti-Stokes Raman scattering (CARS)spectroscopic microscopy. An apparatus for coherent anti-Stokes Raman(CARS) microscopy can include any suitable optical arrangement as shownin FIGS. 1, 3, 5A-5E, 12A, 12B 13A-13C, 18, 19, and 24. In particular,the apparatus for CARS microscopy can include an object plane 122, 222,1222, 1822 on which can be disposed a first surface of a planarsubstrate 125, 225, 1225, 1825 or plane-parallel optical element 2405,wherein the substrate 125, 225, 1225, 1825 can be characterized by ahomogeneous refractive index (n_(sub)) and a surface normal 226, 1226,1826 and plane-plane parallel optical element 2405 can be characterizedby a homogeneous refractive index (n_(pp)) and a surface normal 2406.The apparatus for CARS microscopy can also include a first opticalsystem disposed to provide a illumination of the object plane 122, 222,1222, 1822, the Illumination characterized by two substantiallycoincident coherent beams 110, 110′, 210, 210′, with wavelengths λ₁ andλ₂ and corresponding angular frequencies ω₁ and ω₂ with ω₁>ω₂, a radiusof curvature, and disposed at one of a plurality of incident wavevectors from about 0 to about 2πn_(sub)/λ₁ or 2πn_(pp)/λ₁ with respectto a surface normal of the substrate 125, 225, 1225, 1825 orplane-parallel optical element 2405 and at a multiplicity of azimuthangles spanning 0 to 2π. The apparatus for CARS microscopy can alsoinclude a second optical system (collection) 130, 230, 530, 1230, 1830,or 2420A-C having an optical axis 136, 236, 536, 1236, 1836, 2406disposed at one of a plurality of center wave vectors from about 0 toabout 2πn_(sub)/λ₁ or 2πn_(pp)/λ₁ with respect to the surface normal,wherein the second optical system 130, 230, 530, 1230, 1830, 2420A-C ischaracterized by a numerical aperture (NA) or NA_(pp) and is responsiveprimarily to optical signals at frequencies greater than ω₁. Theapparatus for CARS microscopy can further include a third optical systemdisposed in an optical path of the first optical system to provideinterferometric reintroduction of a reference illumination (referencebeam) at a frequency of 2ω₁−ω₂, into the second optical system 130, 230,530, 1230, 1830, 2420A-C wherein each of an amplitude, a phase, a radiusof curvature and an angle of incidence of the reference is adjusted asrequired such that a corrected reference wave is present at the imageplane of the second optical system 130, 230, 530, 1230, 1830, 2420A-C.The apparatus for CARS microscopy can also include an electronic imagedevice disposed at an image plane 124, 224 of the second optical system130, 230, 530, 1230, 1830, 2420A-C that responds linearly to the localoptical intensity and transfers the local optical intensity map acrossthe image plane (a sub-image) to a signal processor device in electronicform, a signal processor that receives the electronic form of thesub-image and manipulates the sub-image to correct for distortions andalteration introduced by the optical configuration, and an electronicdevice to sequentially collect, store and combine a plurality ofsub-images corresponding to a plurality of regions of spatial frequencyspace to create a composite image, wherein the plurality of sub-imagesare formed as a result of adjustments to the first, the second, and thethird optical systems.

In various embodiments, the third optical system of the apparatus forCARS microscopy can include a first beamsplitter disposed in the opticalpath of the first optical system before the object plane 122, 222, 1222,1822, 2425 to collect a portion of the coherent illumination and one ormore optics disposed between the first optical system and the secondoptical system 130, 230, 530, 1230, 1830, 2420A-C wherein the opticsincludes a nonresonant nonlinear material configured to generate theanti-Stokes four-wave mixing frequency 2ω₁−ω₂ and exclude thefundamental frequencies (ω₁ and ω₂), and to interferometricallyreintroduce the portion of the anti-Stokes coherent illumination as areference beam into the second optical system 130, 230, 530, 1230, 1830,2420A-C in a position after the exit aperture of a collection(objective) lens, wherein the reintroduction is at one of a positioncorresponding to a position a zero-order beam would have had if it hadbeen transmitted through an appropriate higher NA lens of the secondoptical system 130, 230, 530, 1230, 1830, 2420A-C as shown in FIG. 1 oran aliased position to reduce pixel requirements of the electronic imagedevice, wherein the signal processor is adjusted to compensate for thisspatial frequency aliasing.

In various embodiments, the third optical system of the apparatus forCARS microscopy can include one of the third optical systemconfigurations shown in FIGS. 5A-5E. In some embodiments, the apparatusfor CARS microscopy can include a third optical system 500E in aconfiguration shown in FIG. 5E. The third optical system can include afirst beamsplitter disposed in the optical path of the first opticalsystem before the object plane 522 to collect a portion of the coherentillumination one or more transfer optics disposed between the firstoptical system and the second optical system 530, wherein the opticsincludes a nonresonant nonlinear material 520 configured to generate theanti-Stokes four-wave mixing frequency 2ω₁−ω₂ and exclude thefundamental frequencies (ω₁ and ω₂), and a second beamsplitter 570disposed between the object plane 522 and a front aperture of acollection lens (objective) of the second optical system 530 toreintroduce the portion of the anti-Stokes coherent wave illumination asa reference beam 510′ into the second optical system 530 at an angle θless than the entrance angular aperture (<˜sin⁻¹ NA) of the secondoptical system 530.

In other embodiments, the apparatus for CARS microscopy can include athird optical system 500D in a configuration shown in FIG. 5D. The thirdoptical system 500D can further include a first beamsplitter disposed inthe optical path of the first optical system to collect a portion of thecoherent illumination, one or more transfer optics disposed between thefirst optical system and the second optical system, wherein the opticsincludes a nonresonant nonlinear material configured to generate theanti-Stokes four-wave mixing frequency 2ω₁−ω₂ and exclude thefundamental frequencies (ω₁ and ω₂). The third optical system 500D canalso include at least one of a grating 584 or a grating on a waveguidedisposed between the object plane 522 and a front aperture of thecollection lens (objective) of the second optical system 530 toreintroduce the portion of the anti-Stokes coherent wave illumination asa reference beam 510′ into the second optical system 530 at an angle θless than the entrance angular aperture (<˜sin⁻¹ NA) of the secondoptical system 530.

In other embodiments, the apparatus for CARS microscopy can include athird optical system 500A in a configuration shown in FIG. 5A. The thirdoptical system 500A can further include a first beamsplitter disposed inthe optical path of the first optical system to collect a portion of thecoherent illumination, one or more transfer optics, wherein the one ormore optics can include a nonresonant nonlinear material configured togenerate the anti-Stokes four-wave mixing frequency 2ω₁−ω₂ and excludethe fundamental frequencies (ω₁ and ω₂) and wherein at least one of theone or more optics is disposed to direct the portion of the anti-Stokescoherent plane wave illumination as a reference beam to illuminate theobject at an angle θ corresponding to less than the entrance angularaperture (<˜sin⁻¹ NA) of the second optical system 530. The thirdoptical system 500A can also include a dynamic (on/off) physical block550 disposed in a back pupil plane of the second optical system 530 toalternately block and unblock a small portion of the pupil aperturecorresponding to the position of the reference beam 510′ in theaperture.

In various embodiments, the apparatus for CARS microscopy can include athird optical system 500C in a configuration shown in FIG. 5C. The thirdoptical system 500C can further include a first beamsplitter disposed inthe optical path of the first optical system to collect a portion of thecoherent illumination, one or more transfer optics, wherein the one ormore optics can include a nonresonant nonlinear material configured togenerate the anti-Stokes four-wave mixing frequency 2ω₁−ω₂ and excludethe fundamental frequencies (ω₁ and ω₂) and wherein at least one of theone or more optics is disposed to direct the portion of the anti-Stokescoherent plane wave illumination as a reference beam to illuminate theobject at an angle θ corresponding to less than the entrance angularaperture (<˜sin⁻¹ NA) of the second optical system 530. The thirdoptical system 500C can also include a guided-mode resonance filter(k-vector filter) 582 disposed between the object plane 522 and acollection lens of the second optical system 530 and an another device(not shown) to adjust the position, tilt and rotation of the guided-moderesonance filter 582 between positions, tilts and rotations in which italternately transmits and blocks the portion of the reference beamtransmitted through the object plane.

In certain embodiments, the apparatus for CARS microscopy can alsoinclude at least one known reference object to cover a small part of theimage field. In some embodiments, the first, the second, and the thirdoptical systems can be arranged in a transmission configuration.

In other embodiments, the first, the second, and the third opticalsystems can be arranged in a reflection configuration. In someembodiments, the plurality of incident wave vectors of the first opticalsystem can include wave vectors less than about 2π/λ₁ wherein these wavevectors are accessed by illumination of the substrate at polar anglesbetween 0 and π/2. In other embodiments, the plurality of incident wavevectors of the first optical system can include wave vectors betweenabout 2π/λ₁ and about 2πn_(sub)/λ₁ or 2πn_(pp)/λ₁, wherein these wavevectors are accessed by evanescent wave illumination of the objectthrough the substrate. Furthermore, the apparatus for CARS microscopycan use any of the arrangements shown in FIGS. 13A-13C for couplinglight into the substrate for illumination through the substrate 125,225, 1225, and 1825 or plane-parallel optical element 2405.

In some other embodiments, the plurality of center wave vectors of thesecond optical system 130, 230, 530, 1230, 1830, 2420A-C can includeonly center wave vectors less than about 2π/λ₁, wherein these centerwave vectors are accessed by an optical system above the object plane ofthe substrate 125, 225, 1225, 1825 or plane-parallel optical element2405. In certain embodiments, the plurality of center wave vectors ofthe second optical system 130, 230, 530, 1230, 1830, 2420A-C can includecenter wave vectors between 2π/λ₁ and 2πn_(sub)/λ₁ or 2πn_(pp)/λ₁,wherein the center wave vectors greater than 2π/λ₁ are accessed throughthe substrate 125, 225, 1225, 1825 or plane-parallel optical element2405 and the second optical system 130, 230, 530, 1230, 1830, 2420A-Ccan include one or more gratings on the side of the planar substrate125, 225, 1225, 1825 or one or more gratings 2415 on the side of theplane-parallel optical element 2405 opposite the object plane 122, 222,1222, 1822, 2425 wherein each grating is characterized by a position, apitch, and a grating profile.

According to various embodiments, there is a method for coherentanti-Stokes Raman (CARS) microscopy. The method for CARS microscopy caninclude providing an object 120, 220, 1220, 1820, 2410 disposed over aplanar substrate 125, 225, 1225, 1825 or plane-parallel optical element2405, wherein the substrate 125, 225, 1225, 1825 is characterized by ahomogeneous refractive index (n_(sub)) and a surface normal and whereinthe plane-parallel optical element 2405 is characterized by ahomogeneous refractive index (n_(pp)) and a surface normal and providinga first optical system disposed to provide a illumination of the objectplane 122, 222, 1222, 1822, 2425 the illumination characterized by twosubstantially coincident coherent beams with wavelengths λ₁ and λ₂ andcorresponding angular frequencies ω₁ and ω₂ with ω₁>ω₂, a radius ofcurvature, and disposed at one of a plurality of incident wave vectorsfrom about 0 to about 2πn_(sub)/λ₁ or 2πn_(pp)/λ₁, with respect to asurface normal 126, 226, 12268, 1826, 2406 of the substrate 125, 225,1225, 1825 or plane-parallel optical element 2405 and at a multiplicityof azimuth angles spanning 0 to 2π. The method can also includeproviding a second optical system (collection) 130, 230, 1230, 1830,2420A-C having an optical axis 136, 236, 1236, 1836, 2406 disposed atone of a plurality of center wave vectors from about 0 to about2πn_(sub)/λ₁ or 2πn_(pp)/λ₁ with respect to the surface normal 125, 225,1225, 1825, 2406 wherein the second optical system 130, 230, 1230, 1830,2420A-C is characterized by a numerical aperture (NA) and is responsiveprimarily to optical signals at frequencies greater than ω₁ andproviding a third optical system disposed in an optical path of thefirst optical system to provide interferometric reintroduction of areference illumination (reference beam) at a frequency of 2ω₁−ω₂, intothe second optical system 130, 230, 1230, 1830, 2420A-C wherein each ofan amplitude, a phase, a radius of curvature and an angle of incidenceof the reference is adjusted as required such that a corrected referencewave is present at the image plane of the second optical system 130,230, 1230, 1830, 2420A-C. The method can further include providing anelectronic image device disposed at an image plane of the second opticalsystem 130, 230, 1230, 1830, 2420A-C that responds linearly to the localoptical intensity and transfers the local optical intensity map acrossthe image plane (a sub-image) to a signal processor device in electronicform, providing a signal processor that receives the electronic form ofthe sub-image, manipulating the sub-image using the signal processor tocorrect for distortions and alteration introduced by the opticalconfiguration, providing an electronic device to sequentially collect,store and combine a plurality of sub-images corresponding to a pluralityof regions of spatial frequency space to create a composite image,wherein the plurality of sub-images are formed as a result ofadjustments to the first, the second, and the third optical systems, andcombining the plurality of sub-images into a composite image to providea substantially faithful image of the object 120, 220,1220, 1820, 2410.

According to various embodiments, the method can further include one ormore processes of subtraction of dark field images, subtraction ofbackground images, shifting of spatial frequencies in accordance withthe optical configuration, and elimination of one or more overlappingcoverages of the frequency space wherein the elimination operations canbe performed either in the optical systems or in the signal processing.In some embodiments, the method can further include selecting regions ofspatial frequency space to provide a more or less faithful image of theobject 120, 220, 1220, 1820, 2410 in the object plane 122, 222, 1222,1822, 2425.

Up to this point, IIM has been discussed in the context of imaging 2Dobjects (e.g., thickness of object <<wavelength, for example, Cr onglass masks) because, at a single frequency, scattering from multipleobjects in multiple z-planes (displaced along the optical axis of theobjective) all contribute to the image and make imaging of 3D objectsproblematic. However, 3D imaging is necessary to observe a variety ofimportant objects, including a majority of biological moieties. Inconventional IIM, which is a single-side-band system with a relativelylong coherence length, it is difficult, if not impossible, to recordseparate information from a thin in-focus layer without severecontamination by information from other (defocused) layers. This problemcan be solved by using multiple angles of incidence and multiplewavelengths.

To start the discussion, the following description presents an exemplarymodel for sectioning of just two longitudinally separated 2D-objectsusing illuminations with two different wavelengths. Following thatdiscussion, an algorithm is presented to extend to multiple image planesand thereby to 3D imaging.

FIG. 36 shows an example configuration for sectioning a 3D object, wherethe 3D object is represented by substantially planar object-A 3605displaced from substantially planar object-B 3610. In this example,object-A 3605 is positioned on a first surface 3615 of a substrate 3620and displaced by a predetermined thickness of the substrate 3620 fromobject-B 3510 that is positioned on the opposite surface 3625 of thesubstantially plane parallel substrate 3620 characterized by arefractive index n_(pp) and a thickness Δz. For simplicity, we taken_(pp)=n_(med), the index of the surrounding medium (usually air) tosimplify the formulas, this eliminates refraction effects at theboundary between the media as well as aberrations associated withimaging through layered inhomogeneous media. With these simplifications,the sample consists of two 2D objects in two parallel planes separatedby a distance Δz immersed in a homogeneous medium. The normal to the twoplanes is chosen as the longitudinal axis of the composite (two plane)object. Collimated radiation (the first optical system) with a firstwavelength λ^(j) 3630 and a second wavelength λ^(j′) 3635 is directed toilluminate both object-A 3605 and object-B 3610, and the second(collection) optical system 3640 is adjusted such that object-A 3605 isin focus and for Δz greater than the depth of field of the secondoptical system, object-B 3610 is out of focus. In this example, bothobject-A 3605 and object-B 3610 are nested “ell's”, but object-B 3610has slightly different pitch and is rotated 180° relative to object-A3605 for ease of identification. Focusing on the object-A 3605 and usingwavelength λ_(j), both a focused image of object-A and a defocused imagefor object-B 3610 is recorded by the collection optical systemrepresented in the figure by collection optic 3640.

In this example and for simplicity, weakly-scattering objects areassumed, which means that scattering from both objects does notsignificantly perturb the incident light and multiple scatteringinvolving both object planes is negligible. This approximation issuitable for many objects and in particular for biological samples whichare generally weak-scatterers.

In general the optical arrangement can be manipulated so that theillumination and collection directions each can take any directionwithin 4π steradians. That is each of the illumination direction and theoptical axis of the collection system and be adjusted arbitrarilyrelative to the object. Thus, four angles (two polar angles and twoazimuthal angles are required to describe the arrangement. Without lossof generality, the collection azimuthal angle can be fixed in a specificplane of the laboratory system and the sample can be allowed to rotateabout its longitudinal axis. Thus the four independently adjustableangles are the polar and azimuthal angles of the illumination and therotation of the sample (first optical system: θ_(ill), φ_(ill), andφ_(r)) and the tilt of the collection system (second optical system:θ_(tilt)). We further track φ_(ill) and φ_(r) so that the offsetfrequencies due to the illumination are independent of the rotation ofthe sample. This choice simplifies the resulting equations but does notin any way restrict the geometry. The choice of rotating the object orrotating the collection system will be dictated by detailed experimentalconsiderations. Mathematically, they are equivalent.

Since the illumination direction is now specified by two angles a polarangle and an azimuthal angle, there are spatial frequency offsets inboth the x and y directions, viz:

$\begin{matrix}{{f_{{ill},x}^{j,k} = {\frac{2\pi}{\lambda^{j}}n_{med}{\sin \left( \theta_{ill}^{k} \right)}{\cos \left( \phi_{ill}^{k} \right)}}}{f_{{ill},y}^{j,k} = {\frac{2\pi}{\lambda^{j}}n_{med}{\sin \left( \theta_{ill}^{k} \right)}{\sin \left( \phi_{ill}^{k} \right)}}}} & (25)\end{matrix}$

where the superscripts refer to the j'th wavelength and the k'th pair ofoffset angles. The (x,y) subscripts refer to the directions in theobject frame. Since electromagnetic fields are vector fields, it isnecessary to track the polarization directions which in general becomequite complex. For simplicity in this treatment, we take a scalarapproach that is approximately correct only for small illuminationangles, θ_(ill). In an actual imaging application the vector relations,which impact the interferometric amplitudes, but not the phases, willhave to be retained.

If we now add a rotation by an angle φ_(r) about the object longitudinal(z-axis) from the (x,y) coordinates to (x′,y′) coordinates, we have asimple rotation transformation:

$\begin{matrix}{{{\begin{pmatrix}f_{q^{\prime}} \\f_{g^{\prime}}\end{pmatrix} = {\begin{pmatrix}{\cos \left( \phi_{r} \right)} & {- {\sin \left( \phi_{r} \right)}} \\{\sin \left( \phi_{r} \right)} & {\cos \left( \phi_{r} \right)}\end{pmatrix}\begin{pmatrix}f_{q} \\f_{g}\end{pmatrix}}};}{\begin{pmatrix}f_{q} \\f_{g}\end{pmatrix} = {\begin{pmatrix}{\cos \left( \phi_{r} \right)} & {\sin \left( \phi_{r} \right)} \\{- {\sin \left( \phi_{r} \right)}} & {\cos \left( \phi_{r} \right)}\end{pmatrix}{\begin{pmatrix}f_{q^{\prime}} \\f_{g^{\prime}}\end{pmatrix}.}}}} & (26)\end{matrix}$

where (f_(q),f_(g)), (f_(q′),f_(g′)) correspond to the spatial frequencycomponents of the image in the (x,y) and (x′,y′) coordinate systems,respectively. We will take the frequencies as f_(q)=qf_(x) andf_(g)=gf_(y) where f_(x)=2π/L_(x) and f_(y)=2π/L_(y), and L_(x) andL_(y), roughly set by the dimensions of the field-of-view of the secondoptical system, refer to the lowest nonzero spatial frequencies in theFourier series expansion as described in connection with Eq. (1) and qand g are integers specifying the harmonics of these basis frequenciesin the (x,y) object plane and similar expressions f_(q′) and f_(g′),with

f _(x′)=√{square root over ((f _(x) cos φ_(r))²+(f _(y) sin φ_(r))²)}; f_(y)=√{square root over ((f _(x) sin φ_(r))²+(f _(y) cosφ_(r))²)}.  (27)

Relationship Eq. (26) is an analog equation while the frequencies aredigitally indexed (q,g) and (q′,g′). Throughout this discussion we usethe simple digitization procedure of taking the closest integer to theanalog value. This introduces some distortions into the images which canbe made smaller by taking a finer frequency grid, at the expense ofincreased computation time. The digital signal processing community hasdealt with this issue at great length and multiple approaches areavailable.

Under these conditions, each spatial Fourier component

_(qp) of the total image

can be described as:

$\begin{matrix}{{{\;}_{q\; g}^{j,k} = {{\;}_{q\; g} + {{\;}_{q\; g}\; \text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (28)\end{matrix}$

here

_(qg) and

_(qg) are spatial Fourier coefficients of the original objects in thesense of Eq. (5) and φ_(qg) ^(j,k)=Δz(−r_(j,k)+s_(qg) ^(j,k)) is theincrement in phase of the spectral component

_(qg) ^(j) occurring as a result of the separation Δz. Here

$\begin{matrix}{{r^{j,k} = {\sqrt{\left( {\frac{2\pi}{\lambda^{j}}n_{med}} \right)^{2} - \left( f_{{ill},x}^{j,k} \right)^{2} - \left( f_{{ill},y}^{j,k} \right)^{2}}\left( {= {\frac{2\pi}{\lambda^{j}}n_{med}{for}\mspace{14mu} {normal}\mspace{14mu} {incidence}\mspace{14mu} {illumination}}} \right)}};} & (29) \\{\mspace{79mu} {{s_{qg}^{j,k} \equiv \left\{ \sqrt{\left( {\frac{2\pi}{\lambda^{j}}n_{med}} \right)^{2} - \left( {{qf}_{x} - f_{{ill},x}^{j,k}} \right)^{2} - \left( {{gf}_{y} - f_{{ill},y}^{j,k}} \right)^{2}} \right\}},}} & (30)\end{matrix}$

The phase shifts resulting from the illumination (the first opticalsystem, e.g. the wavelengths λ^(j) (j=1, 2, . . . ) and the anglesθ_(ill) and φ_(ill)) is independent of the second optical systemconfiguration; these phase shifts provide the necessary information tounravel the 3D images from the measured convoluted images.

In this section for clarity, we have adopted the notation thatsubscripts refer to the scattering components corresponding to differentspatial frequencies and the superscripts refer to (first index—j) themeasurement wavelength, (second index—k) the illumination configurationdefined by the first optical system (with normal incidence illuminationgiven the index 0 and off-axis illumination having progressively higherindices, and (third index—l) denoting the configuration (tilt) of thesecond optical system. Note that the absence of superscripts on (q,g)implies that these coefficients are independent of wavelength—so that(q,g) refer to the same spatial frequencies independent of theillumination wavelength and the optical arrangements. Therefore, as theincident offsets, f_(ill,x) ^(j,k) and f_(ill,y) ^(j,k) and the rotationof the sample are varied, the directions of propagation of the planewaves corresponding to the spatial frequencies indexed to q,g are variedas well. Note that we take the illumination system as rotating alongwith the sample, this does not in any way restrict the available anglesbut simplifies the ‘book-keeping’ of the observed spatial frequencies;in particular with this convention, the phase shifts between differentconfigurations and wavelengths are independent of rotation. However, therotation does allow collection of additional spatial frequencyscattering components. Changing the illumination wavelength will alsochange the wavevector and hence the propagation direction of lightscattered by the (q,g) spatial frequency component of the object. If thesecond optical system is changed by tilting the optical axis, thelaboratory frame frequencies are nonlinearly mapped into the objectframe frequencies, but the phase shifts are not changed.

The observed image at the camera is described as:

$\begin{matrix}{{\overset{\_}{I}}_{\eta,\gamma} = {\Sigma_{\overset{\sim}{o}}\mspace{11mu} }} & (31)\end{matrix}$

where the bars indicate the camera image plane, (η,γ) are the spatialcoordinates in the camera plane and (õ,{tilde over (p)}) are thecorresponding spatial frequency indices in the camera frame. The factorM accounts for the magnification of the second optical system and thespatial frequencies are measured in the image plane of the secondoptical system.

The spatial frequencies at the camera are the result of interferencebetween the scattered spatial frequency plane waves collected by thesecond optical system (of_(x′), pf_(y′)) referenced to the tiltedoptical axis of the second optical system and the reference beamcharacterized by polar angles θ_(α) and φ_(α) also referenced to thetilted optical axis.

$\begin{matrix}{\begin{pmatrix}{oMf}_{x^{\prime}} \\{pMf}_{y^{\prime}}\end{pmatrix} = {\begin{pmatrix}{\overset{\sim}{o}\; {Mf}_{x^{\prime}}} \\{\overset{\sim}{p}\; {Mf}_{y^{\prime}}}\end{pmatrix} - {\frac{2\pi \; n_{med}}{\lambda^{j}}\begin{bmatrix}\left( {\cos \; \theta_{\alpha}\cos \; \phi_{\alpha}} \right) \\\left( {\cos \; \theta_{\alpha}\sin \; \phi_{\alpha}} \right)\end{bmatrix}}}} & (32)\end{matrix}$

where the phase of the reference wave is set, for example, by comparisonwith a known object, so that the phases of the scattered waves aresimply given by the phases of the plane waves scattered from the objectand the common mode propagation effects are compensated at the cameraimage plane; this is equivalent to setting the origin of the conjugateimage plane of the second optical system to the origin of the objectcoordinate system. In addition to the conversion of this equation fromanalog to digital form discussed above, there is another source ofdigitization error in this result associated with the finite size of thecamera pixels. Again, this is a well studied issue in digital imageprocessing.

It remains to relate the frequencies observed in the laboratory framesub-images to the spatial frequencies (qf_(x), gf_(y)) in the objectplane. The object coordinate system 3701 is rotated sequentially aboutthe z-axis (to access different parts of spatial frequency space, φ_(r))followed by a rotation about the y′-axis (to align the z″ axis with theoptical axis of the second optical system and eliminate θ_(tilt)) at3702 and 3703, respectively, in FIG. 37. It remains to connect thelaboratory frame observed spatial frequency indices (o,p) with theobject plane spatial frequencies (q,g). This is straightforward usingsequentially applied coordinate rotation matrices:

$\begin{matrix}{{\begin{pmatrix}{oMf}_{x^{\prime}} \\{pMf}_{y^{\prime}} \\k_{z^{\prime}}\end{pmatrix} = {{\left( \begin{pmatrix}{\cos \; \theta_{tilt}} & 0 & {\sin \; \theta_{tilt}} \\0 & 1 & 0 \\{{- \sin}\; \theta_{tilt}} & 0 & {\cos \; \theta_{tilt}}\end{pmatrix} \right)\begin{pmatrix}{\cos \; \phi_{r}} & {{- \sin}\; \phi_{r}} & 0 \\{\sin \; \phi_{r}} & {\cos \; \phi_{r}} & 0 \\0 & 0 & 1\end{pmatrix}\begin{pmatrix}{{qf}_{s} - f_{{ill},x}^{j,k}} \\{{gf}_{y} - f_{{ill},y}^{j,k}} \\\sqrt{\left( {2\pi \; {n_{med}/\lambda^{j}}} \right)^{2} - \left( {{qf}_{x} - f_{{ill},x}^{j,k}} \right)^{2} - \left( {{gf}_{y} - f_{{ill},y}^{j,k}} \right)^{2}}\end{pmatrix}} = {\begin{pmatrix}{\cos \; \theta_{tilt}\cos \; \phi_{r}} & {\cos \; \theta_{tilt}\sin \; \phi_{r}} & {\sin \; \theta_{r}} \\{\sin \; \phi_{r}} & {\cos \; \phi_{r}} & 0 \\{\sin \; \theta_{tilt}\cos \; \phi_{r}} & {\sin \; \theta_{tilt}\sin \; \phi_{r}} & {\cos \; \theta_{tilt}}\end{pmatrix}\begin{pmatrix}{{qf}_{x} - f_{{ill},x}^{j,k}} \\{{gf}_{y} - f_{{ill},y}^{j,k}} \\\sqrt{\left( {2\pi \; {n_{med}/\lambda^{j}}} \right)^{2} - \left( {{qf}_{x} - f_{{ill},x}^{j,k}} \right)^{2} - \left( {{gf}_{y} - f_{{ill},y}^{j,k}} \right)^{2}}\end{pmatrix}}}}\mspace{20mu} {and}\mspace{20mu} {k_{z^{\prime}} = {\left( \sqrt{\left( {2\pi \; {n_{med}/\lambda^{j}}} \right)^{2} - \left( {oMf}_{x^{\prime}} \right)^{2} - \left( {pMf}_{y^{\prime}} \right)^{2}} \right).}}} & (33)\end{matrix}$

So:

$\begin{matrix}{{o = {{Integer}\left\{ {\frac{1}{{Mf}_{x^{\prime}}}\begin{bmatrix}{{\cos \; \theta_{tilt}\cos \; {\phi_{r}\left( {{qf}_{x} - f_{{ill},x}^{j,k}} \right)}} - {\cos \; \theta_{tilt}\sin \; {\phi_{r}\left( {{gf}_{x} - f_{{ill},y}^{j,k}} \right)}} -} \\{\sin \; \theta_{tilt}\sqrt{\left( {2\pi \; {n_{med}/\lambda^{j}}} \right) - \left( {{qf}_{x} - f_{{ill},x}^{j,k}} \right)^{2} - \left( {{gf}_{y} - f_{{ill},y}^{j,k}} \right)^{2}}}\end{bmatrix}} \right\}}},{p = {{Integer}\left\{ {\frac{1}{{Mf}_{y^{\prime}}}\left\lbrack {{\sin \; {\phi_{r\;}\left( {{qf}_{x} - f_{{ill},x}^{j,k}} \right)}} + {\cos \; {\phi_{r}\left( {{gf}_{x} - f_{{ill},y}^{j,k}} \right)}}} \right\rbrack} \right\}}}} & (34)\end{matrix}$

where the Integer operation means rounding to the nearest integer.The inverse relations are:

$\begin{matrix}{\begin{pmatrix}{{qf}_{x} - f_{{ill},x}^{j,k}} \\{{gf}_{y} - f_{{ill},y}^{j,k}} \\k_{z}\end{pmatrix} = {{\begin{pmatrix}{\cos \; \phi_{r}} & {\sin \; \phi_{r}} & 0 \\{{- \sin}\; \phi_{r}} & {\cos \; \phi_{r}} & 0 \\0 & 0 & 1\end{pmatrix}\begin{pmatrix}{\cos \; \theta_{tilt}} & 0 & {{- \sin}\; \theta_{tilt}} \\0 & 1 & 0 \\{\sin \; \theta_{tilt}} & 0 & {\cos \; \theta_{tilt}}\end{pmatrix}\begin{pmatrix}{oMf}_{x^{\prime}} \\{pMf}_{y^{\prime}} \\\sqrt{\left( {2\pi \; {n_{med}/\lambda^{j}}} \right)^{2} - \left( {oMf}_{z^{\prime}} \right)^{2} - \left( {pMf}_{y^{\prime}} \right)^{2}}\end{pmatrix}} = {\begin{pmatrix}{\cos \; \theta_{tilt}\cos \; \phi_{r}} & {\sin \; \phi_{r}} & {\cos \; \theta_{tilt}\sin \; \phi_{r}} \\{\cos \; \theta_{tilt}\sin \; \phi_{r}} & {\cos \; \phi_{r}} & {\sin \; \theta_{tilt}\sin \; \phi_{r}} \\{\sin \; \phi_{r}} & 0 & {\cos \; \phi_{r}}\end{pmatrix}\begin{pmatrix}{oMf}_{x^{\prime}} \\{pMf}_{y^{\prime}} \\\sqrt{\left( {2\pi \; {n_{med}/\lambda^{j}}} \right)^{2} - \left( {oMf}_{x^{\prime}} \right)^{2} - \left( {pMf}_{y^{\prime}} \right)^{2}}\end{pmatrix}}}} & (35) \\{{q = {{Integer}\left\{ {\frac{1}{f_{x}}\begin{bmatrix}{{\cos \; \theta_{tilt}\cos \; {\phi_{r}\left( {oMf}_{x^{\prime}} \right)}} + {\sin \; {\phi_{r}\left( {pMf}_{y^{\prime}} \right)}} -} \\{{\cos \; \phi_{r}\sqrt{\left( {2\pi \; {n_{med}/\lambda^{j}}} \right)^{2} - \left( {oMf}_{x^{\prime}} \right)^{2} - \left( {pMf}_{y^{\prime}} \right)^{2}}} + f_{{ill},x}^{j,k}}\end{bmatrix}} \right\}}}p = {{Integer}\left\{ {\frac{1}{f_{y}}\left. \quad\begin{bmatrix}{{{- \cos}\; \theta_{tilt}\sin \; {\phi_{r}\left( {oMf}_{x^{\prime}} \right)}} + {\cos \; {\phi_{r}\left( {pMf}_{y^{\prime}} \right)}} +} \\{{\cos \; \theta_{tilt}\sin \; \phi_{r}\sqrt{\left( {2\pi \; {n_{med}/\lambda^{j}}} \right)^{2} - \left( {oMf}_{x^{\prime}} \right)^{2} - \left( {pMf}_{y^{\prime}} \right)^{2}}} + f_{{ill},y}^{j,k}}\end{bmatrix} \right\}} \right.}} & (36)\end{matrix}$

Applying this mapping, we can convert the sub-image to the object plane:

$\begin{matrix}{I_{x,y} = {\Sigma_{q}}} & (37)\end{matrix}$

Using a different wavelength λ^(j′), the spatial Fourier coefficients ofthe recorded image

can be described as Eq. 28 for

is repeated for convenience:

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Eq}.\mspace{14mu} 28} \right\rbrack} & \; \\{\mspace{79mu} {{{\;}_{q\; g}^{j,k} = {{\;}_{qg} + {{\;}_{q\; g}\text{?}}}}\mspace{79mu} {{\;}_{q\; g}^{j,k} = {A_{q\; g} + {B_{q\; g}\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (38)\end{matrix}$

Solving this system of equations, the Fourier coefficients of image

and image

can be reconstructed as:

qg = qg j , k - qg j ′ , k  ϕ qg j , k -  ϕ qg j ′ , k   qg = qg j, k   ϕ qg j , k - qg j ′ , k   ϕ qg j , k  ϕ qg j , k -  ϕ qgj ′ , k ( 39 )

Clearly, this reconstruction fails if φ_(qg) ^(j,k)=φ_(qg)^(j′,k′)(modulo 2π) for any (q,g) pair. This discussion has beenpresented in the context of changing the illumination wavelength.However, because the phase shifts, φ_(qg) ^(j,k) vary with both thewavelength (j) and the illumination geometry (k), it is also possible toprovide longitudinal resolution by varying the first optical system(e.g. the illumination angles for a set of specific (q,g). Some cautionis required, not all measurements will be independent, for some thechanges in the wavelength and in the illumination angles will compensateeach other and result in a redundant measurement. Only non-degeneratemeasurements should be included in the analysis.

If we first consider only the case where multiple wavelengths are used,the maximum contrast occurs when the denominator in Eq. (39) is largest,i.e. when the phase difference φ_(qg) ^(j,k)−φ_(qg) ^(j′,k)=π which setsa relationship between the resolution along the propagation direction(Δz) and the wavelength change Δλ^(jj′)=λ^(j)−λ_(j′) as

Δz _(min)˜λ_(j)λ^(j′)/2(n _(med)Δλ^(jj′)).  (40)

Conventional interferometry, for example as used in 1D stage-positionmeasurements in advanced lithographic steppers, is able to divide awavelength into at least 1000 parts, e.g. the resolution is 1000×better, or the spread in wavelengths is 1000× smaller, than the valueprojected above. This of course depends on the signal/noise level of themeasurement and the ability to experimentally eliminate sources of driftsuch as ambient temperate and humidity changes in the optical paths ofthe interferometer. The trade-off between Δλ and resolution will dependon many experimental parameters.

In a model calculation, Eqs. (38)-(39) are applied to a high frequencyIIM image as shown in FIG. 38, where the focused image A is shown inFIG. 38(a), the defocused image B is shown in FIG. 38(b), the simulated‘recorded’ combination of images {

_(qg)+

_(qg) exp(iφ_(qg) ^(j,k))}_(FT) ⁻¹ is shown in FIG. 38(c). There is asimilar set of images for the second illumination wavelength (notshown). High frequency sub-images of the individual objects,

and

after reconstruction are shown in FIG. 38(d) and FIG. 38(e),respectively. Note that the images are rotated by 180° as expected. Herethe notation sub-FT denotes the sub-image spanning the frequency spaceof the optical system for a single image (in this case high frequencycomponents in the x-direction).

The above model is best applied in a noiseless ideal world, however in areal experiment; the subtraction of two almost identical, but noisy,images needs to be considered. It is clear from Eq. (39) that thequality of the separated images will be strongly dependent on theexperimental signal-to-noise ratio.

To demonstrate the ability to account for a defocusing term, a defocusedimage was recorded in the setup shown in FIGS. 40 and 41, and wasrestored electronically. Pictures of recorded defocused high frequencyimage and electronically refocused one with corresponding models andcrosscuts are shown in FIG. 39, where FIG. 39(a) shows the defocusedmodel, FIG. 39(b) shows the defocused experimental result, FIG. 39(c)shows crosscuts of the defocused model (solid line) and the experimentalresult (dotted line), FIG. 39(d) shows the reconstructed model, FIG.39(e) shows the reconstructed experimental result, FIG. 39(f) showscrosscuts of the reconstructed model (solid line) and the experimentalresults (dotted lines).

Rewriting Eqs. (38)-(39) for a general case where P sectioning layersare involved results in the need to record P sub-images at a total of Pdifferent wavelengths and optical arrangements. Eqs. (38) and (39) takethe form of a system of linear equations:

∥Ψ_(qg)∥*

_(qg)=

_(qg)  (41)

where

_(qg) is a vector of P coefficients at a particular spatial frequency(q,g) from the P layers in the object, each longitudinally separated byΔz_(p);

_(qg) is a vector of P coefficients at a particular frequency f_(qg)from the P sub-images, each recorded with a unique combination ofwavelength and configuration of the first and the second opticalsystems; transfer matrix Ψ_(qg) is a P-by-P matrix of defocusingelements corresponding to the longitudinal position of a particularlayer and the phase shift of a particular sub-image defined as:

$\begin{matrix}\left. \Psi_{qg}^{j,k} \middle| {}_{p}{\equiv {^{\frac{2\pi}{\lambda^{j}}{({{- r^{j,b}} + s_{qz}^{j,k}})}\Delta \; Z_{p}}.}} \right. & (42)\end{matrix}$

assuming equal spacing ΔZ_(p)=pΔz. Here the number of independentmeasurements <(j_(max)+k_(max)) is equal to P the number of slices ofthe object. It is important to note that using a plurality ofconfigurations of the first optical system, reduces the required numberof wavelengths and the total wavelength span for a fixed number ofslices.

The formal solution of Eq. (28) is straightforward:

_(qg)=∥Ψ_(qg)∥⁻¹*

_(qg),  (43)

and it is easy to evaluate as long as matrix Ψ_(qg) is well-conditioned(equivalent to the nonvanishing of the denominator in the reconstructionof the two slice case presented above).

Evidently the degree of degeneracy of the matrix Ψ_(qg) is closelyrelated to the magnitude of the difference of the defocusing terms oftwo adjacent separation layers (see the denominator of Eq. 39):

e ^(iφ) ^(qg) ^(j,k) ^(|) ^(p) −e ^(iφ) ^(qg) ^(j,k) ^(|) ^(p+1) =e^(i{tilde over (φ)}) ^(qg) ^(j,k) ^(|p,p+1)(e ^(1Δφ) ^(qg) ^(j,k) ^(|)^(p,p+1) ^(/2) +e ^(−iΔφ) ^(qg) ^(j,k) ^(|) ^(p,p+1) ^(/2))=2ie^(i{tilde over (φ)}) ^(qg) ^(j,k) ^(|) ^(p,p+1) sin(Δφ_(qg)^(j,k)|_(p,p+1)/2).  (44).

where the notation {tilde over (φ)}_(qg) ^(j,k)|_(p,p+1) refers to theaverage phase between the p, and p+1 slices and similarly for the Δφ.

The larger the absolute value of the denominator the more robust thesolution to the Impact of noise on the separation of the images. Themaximum is achieved when Δφ_(qg)=π. For normal incidence illuminationand the collection optical axis aligned along the longitudinal axis ofthe object assuming a small NA, the resolution is given by Eq. (40) andλ^(j) and λ^(j′) span the full range of wavelengths.

Thus, a type of ‘uncertainty’ relation for estimating the optimal rangeof wavelengths for a given axial resolution can be described by.

$\begin{matrix}{{\Delta \; z\; \Delta \; \lambda_{range}} \sim \frac{\lambda^{\max}\lambda^{\min}}{2n_{med}}} & (45)\end{matrix}$

Here Δλ_(range) is the difference of maximal and minimal wavelengthsused in the system (range of wavelengths) used

So, if required resolution is, for example, 120 nm, then the wavelengthrange for the best results is estimated as

$\begin{matrix}{{{\Delta \; \lambda_{range}} \sim \frac{500^{2}}{2 + 120}} = {1041\mspace{14mu} {{nm}.}}} & (46)\end{matrix}$

Note that this is an overestimation since the derivation of Eq. (45)does not include the contributions of varying the first optical systems(represented by the index k in the previous equations).

The range is on the same order as the wavelength range for givenresolution in OCT microscopy, where longitudinal resolution andwavelength range are connected as

${\Delta \; z\; {\Delta\lambda}_{range}} = {2\mspace{11mu} \ln \mspace{11mu} 2{\frac{{\hat{\lambda}}^{z}}{\pi \; n}.}}$

Phase-shift interferometry, wherein the relative phase of the referenceand the illumination beams is varied across a range of π in a pluralityof steps is well known to provide information about the surface figureof an object (e.g. the flatness for example of a window). This conceptcan be added to the techniques discussed herein to add additionalinformation about the z-dependence of the local refractive index.Similarly, many different signal and image processing techniques arewell known and have been applied to similar problems and areincorporated herein.

In implementations, the weakly-scattering restriction can be removed formatrices which include this angular propagation information, since acontribution of multiple scattering, phase change and attenuation byspatial points along the propagation direction can be added for everyfrequency. Thus, objects which are transparent enough to be recordedwithin good signal-to-noise ratio, but which cannot be considered asweakly-scattering; e.g. where multiple scattering has to be considered,can be imaged.

FIGS. 40 and 41, illumination beam 3901 at a first wavelength λ¹ andillumination beam 3902 at a second wavelength λ² are shown directed tooptical elements, e.g., mirrors, 3903 and 3904, respectively. Opticalelements 3903 and 3904, which are a part of the first optical system, asdescribed above with reference to FIG. 3, are arranged such that theillumination beams 3901 and 3902 are directed onto substrate 3905.Optical element 3903 is arranged such that illumination beam 3901 isincident onto an entrance face 3906 of substrate 3905 at an angle α withrespect to a normal 3907 to substrate 3905, which is then refracted toan angle β, which is shallower than angle α, in substrate 3905 andemerges from substrate 3905 at the angle λ. Similarly, illumination beam3902 is directed by optical element 3904 onto substrate 3905 at theangle λ with respect to the normal 3907 to entrance face 3906 ofsubstrate 3905. Illumination beam 3902 is refracted in substrate 3905 ator about the angle β, wherein the difference from the angle β due todispersion as a result of the refractive index of substrate 3905. Object3908 is positioned or mounted near or on the exit face 3909 of substrate3905. Illumination beams 3901 and 3902 are refracted by substrate 3905and scattered by object 3908 along an object plane 3910 at the angle αand is collected by collection objective or lens 3911 arranged on theexit face 3909 of substrate 3905. For example, collection objective orlens 3911 can be arranged at position 3912 to collect radiation at zeroto low scattering angles or positions 3913 and 3914 to collect radiationat higher scattering angles. Optical elements 3903 and 3904 andcollection objective or lens 3911 can be coupled to respective actuatingmounts (not shown) to adjust angles at which the radiation can becollected. As discussed above, the third optical system (not shown) isarranged to provide the reference beam. Although FIGS. 40 and 41 showonly two illumination beams, this present disclosure is not limited tothis exemplary configuration. More than two illumination beams andadditional collection objectives or lenses can be added to thearrangement as shown in FIGS. 40 and 41.

According to the various embodiments, the method can further includecombination of the techniques of CARS and of 3D imaging to provide a 3Dmapping of the CARS signature of the object.

According to the various embodiments, the method can further includecombination of the techniques of structured illumination (aliasing ofspatial frequencies with 3D imaging.

While the invention has been illustrated with respect to one or moreimplementations, alterations and/or modifications can be made to theillustrated examples without departing from the spirit and scope of theappended claims. In addition, while a particular feature of theinvention may have been disclosed with respect to only one of severalimplementations, such feature may be combined with one or more otherfeatures of the other implementations as may be desired and advantageousfor any given or particular function. Furthermore, to the extent thatthe terms “including”, “includes”, “having”, “has”, “with”, or variantsthereof are used in either the detailed description and the claims, suchterms are intended to be inclusive in a manner similar to the term“comprising.” As used herein, the term “one or more of” with respect toa listing of items such as, for example, A and B, means A alone, Balone, or A and B. As used herein, the symbol “n” or “n_(sub)” will meanthe index of refraction of the substrate when used in context to thesubstrate, unless otherwise expressly noted. For example, n_(clad)represents the index of refraction of a cladding.

Other embodiments of the invention will be apparent to those skilled inthe art from consideration of the specification and practice of theinvention disclosed herein. It is intended that the specification andexamples be considered as exemplary only, with a true scope and spiritof the invention being indicated by the following claims.

What is claimed is:
 1. A method for imaging a 3D object immersed in amedium of index of refraction n_(med) comprising: providing a firstoptical system disposed to provide a substantially coherent illuminationto the 3D object, wherein the illumination is characterized by aplurality of wavelengths λ^(j), j=1, 2, . . . m, with λ^(j+1)<λ^(j),wherein the plurality of wavelengths span a wavelength range ofΔλ=λ¹-λ^(m); at each λ^(j) the illumination is characterized by a centerposition, a radius of curvature, a uniform-intensity illuminationdiameter at a plane of the 3D object, and a wavevector wherein thewavevector is disposed at one of a plurality of incident wavevectorsfrom about 0 to about 2πn_(med)/λ^(j), with respect to a longitudinalaxis of the 3D object and at a plurality of azimuth angles spanningabout 0 to 2π; providing a second optical system comprising an opticalimage recording device and one or more additional optical componentswith a numerical aperture NA, the second optical system defining anoptical axis, wherein the optical recording device is operable tocollect at least a portion of the Illumination from the first opticalsystem scattered from the 3D object, wherein the optical axis of thesecond optical system is disposed at one of a plurality of anglesbetween 0 and π/2 with respect to the longitudinal axis of the 3D objectand at a plurality of azimuth angles spanning about 0 to 2π, wherein thefield-of-view of the second optical system is within a spatial extent ofthe uniform-intensity illumination provided by the first optical system;providing a third optical system disposed in an optical path of thefirst optical system to provide interferometric reintroduction of aportion of the coherent illumination at each λ^(j) as a reference beaminto the second optical system, wherein each of an amplitude, a phase, aradius of curvature, a path length, and an angle of incidence of thereference beam is adjustable such that a reference illumination suitablefor interfering with a portion of the illumination scattered by the 3Dobject and collected by the second optical system is present at an inputof the optical image recording device; recording a plurality ofsub-images of the 3D object at the optical image recording device, oneat each λ^(j), wherein each sub-image is formed as a result ofinterference between scattering resulting from the coherent illuminationof the 3D object and the reference beam; adjusting the first, the secondand the third optical systems to collect a plurality of sub-imagescorresponding to the plurality of wavelengths, to a plurality ofoff-axis illumination conditions, and additionally to a plurality ofdirections of the optical axis of the second optical system with respectto the longitudinal axis of the 3D object; and combining the pluralityof sub-images into a separate composite images of the 3D object.
 2. Themethod of claim 1, further comprising translating a center of afield-of-view of the second optical system relative to a center positionof an illumination spatial extent provided by the first optical system,to extend an area of the 3D image.
 3. The method of claim 1, wherein the3D object comprises two substantially 2D objects separated from eachother with a plane-parallel-bounded homogenous medium characterized by athickness and an index of refraction and wherein the plurality ofwavelengths is reduced to two, λ¹ and λ², and the longitudinal axis isdefined as a normal to the plane-parallel-bounded homogenous medium. 4.The method of claim 1, further comprising: providing a body composed ofa homogeneous medium of index of refraction n_(pp) greater than n_(med)within which the 3D object is immersed and having a plane exit face as afinal surface of the first optical system; locating the 3D object at adistance less than λ_(avg) from the plane exit face of the body;providing for coupling of the coherent illumination to the body by oneof side-coupling, prism coupling an addition of a grating to a face ofthe body opposite the exit face; and whereby the illumination providedby the first optical system is at a wavevector larger than2πn_(med)/λ^(j) and less than 2πn_(pp)/λ^(j) and is an evanescent waveextending from the plane exit face of the body.
 5. The method of claim1, further comprising: providing a plane-parallel-bounded body composedof a homogeneous medium of index of refraction n_(pp) greater thann_(med) and a plane exit face as a final element of the first opticalsystem; providing for coupling of the coherent illumination to the bodyby addition of a grating to the face of the plane-parallel-bounded bodyopposite the exit face; locating the 3D object at a distance less thanλ_(avg) from the plane exit face of the plane-parallel body; whereby theillumination provided by the first optical system is at a wavevectorlarger than 2πn_(med)/λ^(j) and less than 2πn_(pp)/λ^(j) and is anevanescent wave extending from the plane exit face of the plane-parallelbody; and adjusting the second optical system to collect illuminationscattered by the 3D object from the illumination provided by the firstoptical system wherein the illumination that is scattered by the 3Dobject is at a wavevector between 2πn_(med)/λ^(j) and 2πn_(pp)/λ^(j) andis evanescently coupled into the plane-parallel-bounded body and iscoupled out of the plane-parallel-bounded body by a grating on the planeexit face of the plane-parallel-bounded body opposite the 3D object. 6.The method of claim 1, wherein providing the third optical systemfurther comprises: collecting a portion of the coherent illumination ateach λ^(j) by splitting the coherent illumination using a beam splitterdisposed in an optical path of the first optical system, andinterferometrically reintroducing the portion of the coherentillumination as a reference beam after an exit aperture of a collectionlens of the second optical system, wherein the reintroduction is at oneof a position, an amplitude, a phase, a radius of curvature, a pathlength, and an angle of incidence into the third optical system suchthat a sub-image is formed with spatial frequency content that isdirectly related to a spatial frequency content of the Illumination thatis scattered by the 3D object.
 7. The method of claim 1, whereinproviding the third optical system further comprises: collecting aportion of the coherent illumination at each λ^(j) by splitting thecoherent illumination using a first beam combining device disposed in anoptical path of the first optical system, and interferometricallyreintroducing the portion of the coherent illumination as a referencebeam before an entrance aperture of a collection lens of the secondoptical system, wherein the reintroduction is at an angle less thansin⁻¹(NA) of the collection lens, wherein the second beam combiningdevice is selected from a group consisting of: a beamsplitter, a gratingcoupler, and a waveguide filter such that a sub-image is formed withspatial frequency content that is directly related to a spatialfrequency content of the illumination that is scattered by the 3Dobject.
 8. The method of claim 1, further comprising obtainingadditional sub-images by adjusting the phase of the reference beamprovided by the third optical system at the optical image recordingdevice relative to a phase of the illumination provided by the firstoptical system at the 3D object.
 9. The method of claim 1, furthercomprising computationally manipulating each of the sub-images tocorrect for distortions, spatial frequency aliasing, and alterationsintroduced by arrangements of the first, second, and third opticalsystems.
 10. The method of claim 1, wherein the illumination comprisescombinations of two wavelengths (λ^(j) and λ^(j′)) and the methodfurther comprises detecting at an anti-Stokes wavelength[λ^(j)λ^(j′)/(2λ^(j)−λ^(j′))] and tuning a difference between the twowavelengths to obtain a coherent anti-Stokes Raman signature of the 3Dobject.
 11. An apparatus for imaging a 3D object immersed in a medium ofindex of refraction n_(med) with a thickness larger than opticalwavelengths in the medium used for the Imaging, comprising: a mechanicalmechanism to support the 3D object; a first optical system disposed toprovide a substantially coherent illumination to the 3D object, whereinthe illumination is characterized by a plurality of wavelengths λ^(j),j=1, 2, . . . m, with λ^(j+1)<λ^(j), wherein the plurality ofwavelengths span a wavelength range of Δλ=λ¹-λ^(m); at each λ^(j) theillumination is characterized by a center position, a radius ofcurvature, an uniform-intensity illumination diameter at a plane of the3D object, and a wavevector wherein the wavevector is disposed at one ofa plurality of incident wavevectors from about 0 to about2πn_(med)/λ^(j) with respect to a longitudinal axis of the 3D object andat a plurality of azimuth angles spanning about 0 to 2π; a secondoptical system comprising an optical image recording device and one ormore additional optical components characterized by a numerical apertureNA, the second optical system defining an optical axis, wherein theoptical recording device is operable to collect at least a portion ofthe illumination from the first optical system scattered from 3D object,wherein the optical axis of the second optical system is disposed at oneof a plurality of angles between 0 and π/2 with respect to thelongitudinal axis of the object and at a plurality of azimuthal anglesspanning about 0 to 2π, wherein the field-of-view of the second opticalsystem is within a spatial extent of the uniform-intensity illuminationprovided by the first optical system; a third optical system disposed inan optical path of the first optical system to provide interferometricreintroduction of a portion of the coherent illumination at each λ^(j)as a reference beam into the second optical system, wherein each of anamplitude, a phase, a radius of curvature, a path length, and an angleof incidence of the reference beam is adjustable such that a referenceillumination suitable for interfering with the portion of theillumination scattered by the 3D object and collected by the secondoptical system is present at an input of the optical image recordingdevice; the image recording device wherein each sub-image formed as aresult of interference between the illumination that is scattered by the3D object and the reference beam at each λ^(j) is recorded; anadjustment mechanism operable to configure the first, the second, andthe third optical systems to collect a plurality of sub-imagescorresponding to the plurality of wavelengths, to a plurality ofillumination and additionally to a plurality of regions of an objectspatial frequency space; and a signal-processing device operable tocombine the plurality of sub-images into a separate composite image ofthe 3D object.
 12. The apparatus of claim 11, further comprising one ormore optical, mechanical or both optical and mechanical elementsoperable to translate a center of a field-of-view of the second opticalsystem relative to a center position of an illumination spatial extentprovided by the first optical system, to extend an area of the 3D image.13. The apparatus of claim 11, wherein the 3D object comprises of twosubstantially 2D objects separated from each other with aplane-parallel-bounded homogenous medium characterized by a thicknessand an index of refraction and wherein the plurality of wavelengths isreduced to two, λ¹ and λ², and the longitudinal axis is defined as thenormal to the plane-parallel-bounded homogenous medium.
 14. Theapparatus of claim 11, further comprising: a body composed of ahomogeneous medium of index of refraction n_(pp) greater than n_(med)and having a plane exit face as a final surface of the first opticalsystem; and a coupling element operable to couple the coherentillumination to the body by one of side-coupling, prism coupling or anaddition of a grating to a face of the body; wherein the 3D object ispositionable at a distance less than λ_(avg) from the plane exit face ofthe body; whereby the illumination provided by the first optical systemis at a wavevector larger than 2πn_(med)/λ^(j) and less than2πn_(pp)/λ^(j) and is an evanescent wave extending from the plane exitface of the body.
 15. The apparatus of claim 11, further comprising: aplane-parallel-bounded body composed of a homogeneous medium of index ofrefraction n_(pp) greater than n_(med) and a plane exit face as a finalelement of the first optical system; wherein the 3D object ispositionable at a distance less than λ_(avg) from the plane exit face ofthe body; a coupling element operable to couple the coherentillumination into the body by addition of a grating to a face of theplane-parallel-bounded body opposite the exit face; whereby theillumination provided by the first optical system is at a wavevectorlarger than 2πn_(med)/λ^(j) and less than 2πn_(pp)/λ^(j) and is anevanescent wave extending from the plane exit face of the body; and anadjustment element operable to adjust the second optical system tocollect light scattered by the 3D object from the illumination providedby the first optical system wherein the illumination that is scatteredby the 3D object is at a wavevector between 2πn_(med)/λ^(j) and2πn_(pp)/λ^(j) is evanescently coupled into the plane-parallel-boundedbody and is coupled out of the plane-parallel-bounded body by a gratingon the plane exit face of the plane-parallel-bounded body opposite the3D object.
 16. The apparatus of claim 11, wherein the third opticalsystem is further operable to collect a portion of the coherentillumination at each λ^(j) by splitting the coherent illumination usinga beam splitter disposed in an optical path of the first optical system,and interferometrically reintroduce the portion of the coherentillumination as a reference beam after an exit aperture of a collectionlens of the second optical system, wherein the reintroduction is at oneof a position, an amplitude, a phase, a radius of curvature, a pathlength, and an angle of incidence into the third optical system suchthat a sub-image is formed with spatial frequency content that isdirectly related to the spatial frequency content of the illuminationthat is scattered by the 3D object.
 17. The apparatus of claim 11,wherein the third optical system is further operable to collect aportion of the coherent illumination at each λ^(j) by splitting thecoherent illumination using a first beam combining device disposed in anoptical path of the first optical system, and interferometricallyreintroduce the portion of the coherent illumination as a reference beambefore an entrance aperture of a collection lens of the second opticalsystem, wherein the reintroduction is at an angle less than thesin⁻¹(NA) of the collection lens, wherein the second beam combiningdevice is selected from a group consisting of: a beamsplitter, a gratingcoupler, and a waveguide filter such that a sub-image is formed on theoptical image recording device with spatial frequencies directly relatedto spatial frequency content of the illumination that is scattered bythe 3D object.
 18. The apparatus of claim 11, wherein additionalsub-images are obtained by adjusting a phase of the reference beamprovided by the third optical system at the optical image recordingdevice relative to a phase of the illumination beam provided by thefirst optical system at the 3D object.
 19. The apparatus of claim 11,further comprising a signal processing unit comprising a processor and amemory storing one or more algorithms that cause the processor tocomputationally manipulating each of the sub-images to correct fordistortions, spatial frequency aliasing, and alterations introduced bythe combinations of the first, second, and third optical systems. 20.The apparatus of claim 19, further the first optical system is operableto provide illumination with combinations of two wavelengths (λ^(j) andλ^(j′)) and the signal processing unit is for operable to detect at ananti-Stokes wavelength [λ^(j)λ^(j′)/(2λ^(j)−λ^(j′))] and tune thedifference between the two wavelengths to obtain a spatially resolvedcoherent anti-Stokes Raman signature of the 3D object.